Tim Cook was angry.

It was May 2013, and Mr. Cook, the chief executive of Apple, appeared before a United States Senate investigative subcommittee. After a lengthy inquiry, it found that the company had avoided tens of billions of dollars in taxes by shifting profits into Irish subsidiaries that the panel’s chairman called “ghost companies.”

“We pay all the taxes we owe, every single dollar,” Mr. Cook declared at the hearing. “We don’t depend on tax gimmicks,” he went on. “We don’t stash money on some Caribbean island.”

True enough. The island Apple would soon rely on was in the English Channel.

Five months after Mr. Cook’s testimony, Irish officials began to crack down on the tax structure Apple had exploited. So the iPhone maker went hunting for another place to park its profits, newly leaked records show. With help from law firms that specialize in offshore tax shelters, the company canvassed multiple jurisdictions before settling on the small island of Jersey, which typically does not tax corporate income.

Apple has accumulated more than $128 billion in profits offshore, and probably much more, that is untaxed by the United States and hardly touched by any other country. Nearly all of that was generated over the past decade.

The previously undisclosed story of Apple’s search for a new island tax haven and its use of Jersey is among the revelations emerging from a cache of secret corporate records from Appleby, a Bermuda-based law firm that caters to businesses and the wealthy elite.

More from The New York Times:

Broadcom Offers $105 Billion to Buy Rival Chip Maker Qualcomm
Kremlin Cash Behind Billionaire’s Twitter and Facebook Investments
Facebook Says It’s Policing Fake Accounts. But They’re Still Easy to Spot.

The records, shared by the International Consortium of Investigative Journalists with The New York Times and other media partners, were obtained by the German newspaper Süddeutsche Zeitung.

The documents reveal how big law firms help clients weave their way through the gaps between different countries’ tax rules. Appleby clients have transferred trademarks, patent rights and other valuable intangible assets into offshore shell companies, avoiding billions of dollars in taxes. The rights to Nike’s Swoosh trademark, Uber’s taxi-hailing app, Allergan’s Botox patents and Facebook’s social media technology have all resided in shell companies that listed as their headquarters Appleby offices in Bermuda and Grand Cayman, the records show.

“U.S. multinational firms are the global grandmasters of tax avoidance schemes that deplete not just U.S. tax collection but the tax collection of most every large economy in the world,” said Edward Kleinbard, a former corporate tax adviser to such companies, now a law professor at the University of Southern California.

Indeed, tax strategies like the ones used by Apple — as well as Amazon, Google, Starbucks and others — cost governments around the world as much as $240 billion a year in lost revenue, according to a 2015 estimate by the Organization for Economic Cooperation and Development. The group is overseeing efforts to change the global rules that permit such maneuvers.

The disclosures come on the heels of last week’s proposals by Republican lawmakers to provide several new tax benefits for multinational companies, including cutting the federal corporate income tax rate from 35 percent to 20 percent. President Trump has said that American businesses are getting a bad deal under current rules.

But the documents show how major American companies find creative ways to avoid paying anything close to 35 percent.

Apple, for example, pays taxes at a small fraction of that rate on its offshore profits, according to calculations by The Times based on the company’s securities filings. Apple reports that nearly 70 percent of its worldwide profits are earned offshore.

An Apple spokesman, Josh Rosenstock, declined to answer most questions about the company’s tax strategy. He did say that Apple had told regulators — in the United States and Ireland and at the European Commission — about the reorganization of its Irish subsidiaries. “The changes we made did not reduce our tax payments in any country,” he said.

He added: “At Apple we follow the laws, and if the system changes we will comply. We strongly support efforts from the global community toward comprehensive international tax reform and a far simpler system.”

Allergan, Facebook, Nike and Uber said they complied with tax regulations around the world, according to prepared statements.

Congressional Republicans are also seeking to impose a 10 percent tax on some of the profits that American businesses say are earned offshore — half the rate they are proposing for profits in the United States. The lawmakers have also proposed another break, permitting multinationals to bring home more than $2.6 trillion stowed offshore at sharply reduced tax rates. Both proposals, critics say, would only create additional incentives for businesses like Apple to shift more profits into island hideaways.

Appleby is a member of the global network of lawyers, accountants and bankers who set up or manage offshore companies and accounts for clients who want to avoid taxes or keep their finances a secret from authorities, business partners or even spouses. The firm did not respond to questions from The Times about its work for Apple or other companies.

Tax authorities have challenged several of the offshore structures maintained by Appleby and Estera, a spinoff of the law firm’s corporate services business. Nike triumphed over the Internal Revenue Service in a fight over back taxes a year ago; a similar dispute between the I.R.S. and Facebook is continuing.

European regulators are trying to force countries including Ireland, Belgium, Luxembourg and the Netherlands to collect back taxes from big companies that relied on offshore arrangements. Apple is being pursued for $14.5 billion in back taxes after European regulators ruled that its old tax structure amounted to illegal state aid from the Irish government.

Since the mid-1990s, multinationals based in the United States have increasingly shifted profits into offshore tax havens. Indeed, a tiny handful of jurisdictions — mostly Bermuda, Ireland, Luxembourg and the Netherlands — now account for 63 percent of all profits that American multinational companies claim to earn overseas, according to an analysis by Gabriel Zucman, an assistant professor of economics at the University of California, Berkeley. Those destinations hold far less than 1 percent of the world’s population.

Criticism of such profit shifting was largely ignored until government finances around the globe came under pressure in the years following the 2008 financial crisis, when the practice led to government inquiries, tax inspector raids, media scrutiny and promises of reform.

In May 2013, the Senate’s investigative subcommittee released a 142-page report on Apple’s tax avoidance, finding that the company was attributing billions of dollars in profits each year to three Irish subsidiaries that declared “tax residency” nowhere in the world.

Under Irish law, if a company can convince Irish tax authorities that it is “managed and controlled” abroad, it can largely escape Irish income tax. By seeming to run its Irish subsidiaries from its world headquarters in California, Apple ensured that Irish tax residency was avoided.

At the same time, American law dictated that the subsidiaries were only tax residents in the United States if incorporated there. The federal government permits taxes on any income generated by foreign units to be deferred indefinitely, as long as the company says those profits stay offshore.

“Apple has sought the holy grail of tax avoidance: offshore corporations that it argues are not, for tax purposes, resident anywhere in any nation,” then-Senator Carl Levin, Democrat of Michigan, who was the subcommittee chairman, said at the 2013 hearing.

Ireland’s finance minister at the time, Michael Noonan, at first defended his country’s policies: “I do not want to be the whipping boy for some misunderstanding in a hearing in the U.S. Congress.” Ireland had long pursued business-friendly tax policies, which helped lure jobs to the country, primarily for technology and pharmaceutical companies. Apple now has about 6,000 employees in Ireland, including customer service and administrative jobs.

But by October 2013, in response to growing international pressure, Mr. Noonan announced that Irish companies would have to declare “tax residency” somewhere in the world.

At that time, Apple had accumulated $111 billion in offshore cash, mostly in its Irish subsidiaries. Billions of dollars in new profits poured into them each year. Yet they paid almost no corporate income tax.

Company officials wanted to keep it that way. So Apple sought alternatives to the tax arrangement Ireland would soon shut down. And the officials wanted to be quiet about it.

“For those of you who are not aware Apple are extremely sensitive concerning publicity,” wrote Cameron Adderley, global head of Appleby’s corporate department, in a March 20, 2014, email to other senior partners. He added, “They also expect the work that is being done for them only to be discussed amongst personnel who need to know.”

In building Apple’s new tax shelter, Appleby served as something of a general contractor. A key architect was Baker McKenzie, a huge law firm based in Chicago with 77 offices around the world. The firm has a reputation for devising creative offshore structures for multinationals and defending them to tax regulators. It has also fought international proposals for tax avoidance crackdowns.

Baker McKenzie wanted to use a local Appleby office to maintain an offshore arrangement for Apple, its client. For Appleby, Mr. Adderley explained, this potential assignment was “a tremendous opportunity for us to shine on a global basis with Baker McKenzie.”

Baker McKenzie’s San Francisco office emailed a 14-item questionnaire in March 2014 to Appleby’s offices in Bermuda, the British Virgin Islands, the Cayman Islands, Guernsey, the Isle of Man and Jersey.

“Confirm that an Irish company can conduct management activities (such as board meetings, signing of important contracts) without being subject to taxation in your jurisdiction,” the document requested. Baker McKenzie also asked for assurances that the local political climate would remain friendly: “Are there any developments suggesting that the law may change in an unfavourable way in the foreseeable future?”

(A Baker McKenzie spokesman said, “As a matter of general policy, we do not comment on confidential client matters.”)

Apple decided that its new offshore tax structure should use Appleby’s office in Jersey, which is one of the Channel Islands and has strong links to the British banking system. Jersey makes its own laws and is not subject to most European Union legislation, making it a popular tax haven.

But the plan to use Jersey faced a potential snag: In mid-2014, again under pressure from other governments, Irish ministers explored ending a tax shelter known as the “double Irish,” used by scores of companies, including the Appleby clients Allergan and Facebook, as well as Google, LinkedIn and other businesses.

The double Irish allows companies to collect profits through one Irish subsidiary that employs people in Ireland, then route those profits to an Irish mailbox subsidiary that is a tax resident of an offshore haven like Bermuda, Grand Cayman or the Isle of Man.

Irish officials explored a ban on Irish companies claiming tax residency in tax havens. Executives at Allergan — which had used a double Irish for at least a decade, records show — tried to derail the rule change. Terilea Wielenga, then Allergan’s head of tax, was also international president of the Tax Executives Institute, a trade group. She argued to the Irish finance ministry in July 2014 that any such changes should occur slowly.

The campaign seemed to work. “For existing companies, there will be provision for a transition period until the end of 2020,” Mr. Noonan declared in October 2014. The gradual phase-in would apply not just to existing companies but to any new ones created by that December.

This gave Apple just enough time. By the end of the year, Jersey had become the new tax home of the Irish companies Apple Sales International and Apple Operations International.

But a third Apple subsidiary, Apple Operations Europe, instead became resident in Ireland.

Apple would not say why. But tax experts offer one possible reason. While press attention focused on Ireland’s crackdown on the double Irish, officials announced a measure that gave Ireland more appeal: The country expanded its tax deductions for companies that move rights to intellectual property — like patents and trademarks — into Ireland. If an Irish company spent $15 billion buying such rights, even from a fellow subsidiary, it could claim a $1 billion tax deduction each year for 15 years.

Apple declined to say whether it has availed itself of the new benefit.

But J. Richard Harvey, a Villanova law professor and former I.R.S. official who reviewed the Appleby documents, concluded that there was a strong possibility the company moved intellectual property into Ireland to take advantage of the generous tax rules. Based on disclosures in Apple’s American securities filings, he estimated that the transfer was worth about $200 billion.

That would mean that any income that Apple now generates in Ireland could be partially offset by more than $13 billion in tax deductions each year for 15 years.

Apple’s hunt for a tax haven is a familiar tale, said Reuven Avi-Yonah, director of the international tax program at the University of Michigan Law School, who also reviewed the Appleby documents.

“This is how it usually works: You close one tax shelter, and something else opens up,” he said. “It just goes on endlessly.”

Simon Bowers is a reporter for the International Consortium of Investigative Journalists.