Mr. Milner, 55, studied theoretical physics at Moscow State University before moving to the United States, where he attended the Wharton School at the University of Pennsylvania in the early 1990s and then worked for the World Bank in Washington.

He returned to Russia and in the late 1990s and worked as an executive at Bank Menatep, which was founded by Mikhail B. Khodorkovsky, the oil tycoon who was stripped of his company, prosecuted and imprisoned after a televised confrontation with President Vladimir V. Putin.

Mr. Milner eventually teamed with Alisher Usmanov — an Uzbek-Russian oligarch close to the Russian prime minister, Dmitri A. Medvedev — and a former Goldman Sachs executive to build a significant stake in Mail.ru, a Russian internet company that now trades on the London Stock Exchange.

Mr. Milner’s initial American investments came as he served on an innovation commission set up in 2009 by Mr. Medvedev, who was Russia’s president at the time and is something of a tech enthusiast, famously touring Apple’s headquarters in Cupertino, Calif., with Steve Jobs.

In May 2009, Facebook announced an investment of roughly $200 million by Mr. Milner’s company, Digital Sky Technologies, and said the company planned to spend at least $100 million buying additional stock. Eventually, Mr. Milner’s new venture capital firm, DST Global, also amassed a significant stake in Facebook.

“A number of firms approached us, but DST stood out because of the global perspective they bring,” Mark Zuckerberg, Facebook’s chief executive, said at the time.

The documents reviewed by The Times reveal that DST brought something else as well: a connection — through a succession of shell companies — to the Kremlin.

In those years, as part of its diplomatic “reset” with Moscow, the Obama administration was encouraging Russia to learn from the American technology industry. Importing tech knowledge, the theory went, would ease Russia’s dependence on exporting oil and gas.

For the Facebook deal, it was Gazprom, the state-controlled natural-gas giant, that became the bridge. The company, a vital component of the Putin government has employed its financial subsidiary, Gazprom Investholding, to reclaim assets privatized during the 1990s.

Gazprom Investholding is “used for politically important and strategically important deals for the Kremlin,” said Ilya Zaslavskiy, a contributor to the Kleptocracy Initiative, a project of the Hudson Institute, a conservative think tank in Washington. Both VTB and Gazprom Investholding’s parent, Gazprom, are under United States sanctions stemming from Russia’s support of separatists in eastern Ukraine in 2014.

Over several years, Gazprom Investholding and a subsidiary made hundreds of millions of dollars in loans to a company called Kanton Services, according to records from the Panama Papers, the trove of leaked documents from the law firm Mossack Fonseca. Kanton, in turn, owned one of the DST investment vehicles used to buy shares of Facebook. While it is unclear precisely when Kanton first received its stake in the DST entity, Kanton received $197 million of the Gazprom Investholding loans three months before Facebook announced its first deal with Mr. Milner, the records show.

Kanton, based in the British Virgin Islands, had numerous ties to Mr. Milner’s backer, Mr. Usmanov.

Kanton was owned by a longtime Usmanov business associate, and was controlled by Matthias Bolliger, a director of numerous subsidiaries of Mr. Usmanov’s main holding company, USM Holdings, according to an Appleby memo and the Panama Papers. And an email sent by a trust company on the Isle of Man used by Mr. Usmanov referred to Kanton as a “private investment company, (facebook and twitter).”

In the interviews, Mr. Milner said his investors’ identities were not generally public information. And he sought to distance his company from Mr. Usmanov. “I had no knowledge of him using state funds to invest with us,” he said. What’s more, he said, Mr. Usmanov had many potential sources of financing, making it impossible to know whether money from Gazprom Investholding was even used to finance his stake in Facebook.

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To demonstrate this, Mr. Milner set out a series of clear plastic cups with labels like “Alleged Russian government funds,” “Mr. Usmanov” and “Facebook.” He pulled out a large bag of green M&M’s, filled several cups and proceeded to shift the M&M’s from one cup to another. “Money is fungible,” he explained, adding, “You can’t just say that this specific dollar went all the way to Facebook.”

A spokesman for Mr. Usmanov, Rollo Head, did not address specific questions. But in a statement, he insisted that there are clear streams of money that do not mingle. Mr. Usmanov, he said, “did not borrow from or use state or quasi-state funds to make investments in Facebook.”

The Facebook deal was a case study in the way Russia’s oligarchs have mixed public and private roles for their own, and their government’s, benefit: Even as he was investing in Facebook, Mr. Usmanov was general director of Gazprom Investholding.

In fact, Mr. Usmanov had often intertwined his government position with his personal deals, according to a report by the global investigations and security firm Kroll. Kroll described those arrangements as “synergies.”

The Kroll report — a “reputation audit” — had been commissioned by Mr. Usmanov as he set out to burnish his image a year before his deal with Mr. Milner to invest in Facebook. Kroll investigators, relying on public records and interviews, detailed a long and colorful history: time in prison in Uzbekistan (he was later exonerated) and past associations with alleged Russian organized crime figures, according to a draft copy of the report reviewed by The Times.

The investigators also recounted a dizzying number of deals — involving mining, media and technology companies, often with the assistance of the Kremlin and Mr. Medvedev. “Usmanov looks set to end the Putin era as the ‘most improved’ oligarch over the past eight years,” one source told Kroll.

Kroll investigators found that, for some investments, Mr. Usmanov turned to Kanton, the company that would be a part of Mr. Milner’s Facebook investment.

A Facebook spokeswoman, Vanessa Chen, declined to answer specific questions about the deal with DST, calling it a “passive investor” and noting that the company had invested and cashed out several years ago.

Gazprom Investholding called the loans “prudent,” saying they were “provided for general corporate purposes” at above-market rates. The company did not answer a question about what role, if any, Mr. Usmanov played in the loans, except to say that as its general director, he “operated in accordance” with the company’s charter.

To complete the Facebook deal, DST used a pair of lawyers based in Cyprus and Britain who have also set up offshore entities for Russian oligarchs close to Mr. Putin, according to the Panama Papers, United States securities filings and records from Appleby. DST did not answer questions about how it came to use the two lawyers.

After the Facebook deal, Mr. Milner became a fixture in Silicon Valley, spending about $100 million on a 25,000-square-foot house in Los Altos Hills and investing $7 billion in more than 30 companies, including Spotify, Airbnb and Groupon.

In interviews, he has referred to his close relationship to management at the companies in which he invests. Mr. Milner and Mr. Zuckerberg became friends and met monthly. Still, despite his company’s sizable stake, Mr. Milner did not take a board seat or voting rights.

At the 2010 technology conference in San Francisco, Mr. Milner was asked if he provided management with advice along with his cash. “I think it’s actually the opposite,” he replied. “I sometimes get informal advice from them.”

Mr. Milner’s roughly $380 million investment in Twitter was directly backed by another instrument of Kremlin power: Russia’s second-largest bank, VTB.

Sixty-one percent of the bank is owned by the Russian government. VTB’s president, Andrey L. Kostin, is a former Soviet diplomat; Matthias Warnig, on the bank’s supervisory council, is a former East German spy who served in Dresden while Mr. Putin was stationed there with the K.G.B.

VTB has operations across the world, including in the United States. In recent years, it has been involved in a number of politically sensitive deals, including a loan that financed the Russian government’s murky privatization of 19.5 percent of the oil giant Rosneft.

Last year, the Panama Papers revealed that a Cyprus bank owned by VTB was used to transfer hundreds of millions of dollars to accounts connected to Sergei Roldugin, a cellist and close friend of Mr. Putin. It was unknown what the money was for, but some speculated that he was working as a frontman for Kremlin insiders.

Mr. Roldugin did not respond to messages for this article, but said at the time that his wealth came through donations from rich businessmen for the purchase of musical instruments. VTB called the account in the Panama Papers “unsubstantiated.”

“VTB is really a slush fund for Putin,” said Anders Aslund, a resident senior fellow at the Atlantic Council in Washington, and an early economic adviser to the post-Soviet Russian government. The bank, he added, is “the black cashier of the Putin crowd.”

Mr. Milner’s Twitter deal is a complex web of share transfers and offshore financial entities. But its details may offer clues that there was a strategic motive behind VTB’s involvement.

In July 2011, VTB invested at least $191 million in exchange for shares of an Isle of Man company called DST Investments 3, corporate records show. That offshore vehicle was used to buy roughly half of DST Global’s stake in Twitter that month. DST Investments 3 also issued shares to Kanton, the company linked to Mr. Usmanov that was at the center of the Facebook deal.

The Twitter deal had a notable feature: VTB put virtually all of the cash into DST Investments 3, filings show. Kanton contributed almost none. It is highly unusual for investors in DST funds to get stakes without contributing cash, according to a person familiar with the matter.

On May 7, 2014 — six months after Twitter’s initial public offering, when insiders were first permitted to sell their shares — VTB transferred the bulk of its stake in DST Investments 3 to Kanton. DST also cashed out its Twitter investment.

DST had one other investment from VTB, which Mr. Milner compared to a sovereign wealth fund. VTB was an investor in the Chinese internet company JD.com as recently as February 2015, when it transferred its stake to Kanton, Isle of Man records show.

In a statement, VTB said its involvement with Twitter was “solely a financial investment,” sold for a profit, one of several successful deals in the high-tech industry in that period. The bank added, “VTB is a solely commercial bank, we have never had any politically motivated deals.”

Twitter declined to answer a series of questions about VTB, but said that as a matter of policy it had done reviews of all pre-I.P.O. investors.

Mr. Milner would not discuss how VTB came to be an investor, other than to say that the bank helped take Mail.ru public in 2010. According to the Kroll report, the Kremlin had previously ordered VTB to finance other ventures of Mr. Usmanov. The report also details the close relationship between Mr. Usmanov and Mr. Medvedev, finding that Mr. Medvedev helped him win a bid for a lucrative copper ore deposit in Siberia.

Mr. Milner has also been active beyond Silicon Valley. He is a founder of the Breakthrough Prize, a series of lucrative scientific awards. And in July 2015, he was one of several high-profile investors in Cadre, a New York-based real estate technology company founded by Mr. Kushner and his brother, Joshua. (Other investors and partners include Goldman Sachs, George Soros and Peter Thiel’s Founders Fund.)

Mr. Milner said he invested $850,000 in Cadre through a trust, which in turn controls DST Global Advisers, DST Global’s investment management firm. He said he was introduced to the company through an early investor whose name he did not remember. He said he met Jared Kushner only once, at a conference in Aspen, Colo.

Mr. Kushner and Cadre declined to comment.

In his interviews with The Times, Mr. Milner shied away from the controversy now enveloping Facebook and Twitter, his former major investments.

Growing up in the Soviet Union, he said, “if you’re a father and you have a conversation with your son, it’s easy to say, ‘Please don’t get involved in politics.’”

That “was my principle back then and still now,” he said, adding: “I prefer just to focus on business and philanthropy. And this is plenty for me.”

Katie Benner, Conor Dougherty and Andrew E. Kramer contributed reporting. Research was contributed by Frederik Obermaier and Bastian Obermayer of Süddeutsche Zeitung, and by Susan C. Beachy, Kitty Bennett and Doris Burke.