The divisions between Uber and its former chief executive, Travis Kalanick, are widening.

For the past few weeks, Mr. Kalanick, who had resigned as chief executive in June, kept a low profile. Last month, Uber installed a new chief executive, Dara Khosrowshahi, and the company appeared to be trying to get past a turbulent period that included questions about its culture and changes in its top echelons.

But behind the scenes, Mr. Kalanick and Uber’s board continued to wrestle over who had control of the privately held company through the amount of stock they owned and the voting rights that those shares conferred.

On Thursday, Uber and one of its investors, Goldman Sachs, made a proposal to the board that would reduce Mr. Kalanick’s voting power at the company, according to people briefed on the negotiations, who asked to remain anonymous because they were not authorized to speak publicly. The board could vote on the proposal as early as Tuesday, these people said.

In response, Mr. Kalanick made a move late Friday to reassert his control. The former chief, who holds outsize voting rights at Uber, said he had added Ursula Burns, a former chief executive of Xerox, and John Thain, a former chief executive of Merrill Lynch and the New York Stock Exchange, to the eight-member board.

Because of the proposal to reduce voting rights, it is “essential that the full board be in place for proper deliberation to occur,” Mr. Kalanick said in a statement.

In its own statement, Uber said Mr. Kalanick’s move “came as a complete surprise to Uber and its board.” That is why, it added, the company is “working to put in place world-class governance.”

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The moves underscore the increasingly dysfunctional relationship between Uber and Mr. Kalanick, the company’s co-founder. Mr. Kalanick stepped down as chief executive after some of Uber’s investors said he could not remain. Since then, the former chief, who holds a seat on Uber’s board, has battled with other board members, including Benchmark, a venture capital firm that was an early investor in the company.

Benchmark had previously contended that Mr. Kalanick had too much power over Uber and had sued him in an attempt to reduce that control. That suit has been moved to arbitration, allowing Mr. Kalanick to keep his fight with Benchmark — and any potentially damaging disclosures — out of public view. Benchmark declined to comment on Friday.

The back-and-forth also presents a problem for Mr. Khosrowshahi, who has to deal with a deeply divided board. Mr. Khosrowshahi had already had a taste of Uber’s ups and downs in recent days, when the company was told that it would lose its operating license for London, one of the biggest cities where it does business.

The power plays on Uber’s board are centered on a move made by Mr. Kalanick last year that allowed him to obtain outsize control of several board seats. At the time, he got Benchmark to approve an amendment to the company’s charter that gave him the right to nominate three new directors to add to Uber’s eight-member board. Mr. Kalanick occupies one of those seats, and he has contended that he gets the right to fill the other two seats.

To prevent Mr. Kalanick from exercising that right, Uber and Goldman Sachs proposed on Thursday to reduce his voting rights. If approved, the proposal would also reduce voting power for other early Uber shareholders and board members, including Benchmark, Lowercase Capital and Menlo Ventures.

Uber is also negotiating a sale of some of its existing shares to new investors, including the Japanese conglomerate SoftBank. Goldman Sachs is also one of the financial firms that is managing Uber’s potential share sale to SoftBank.

The fight over voting speaks to the balance of power at young Silicon Valley start-ups. In recent years, entrepreneurs have asked for — and been given — more voting rights by venture capitalists and other investors who are eager to get into a hot deal. Other companies, like Snap and Facebook, also have structures that allow their founders to hold disproportionate voting power.

These sorts of bare-knuckle fights usually unfold behind the scenes in venture capital, where investors and founders have incentives to maintain a positive public persona. Entrepreneurs start companies more than once, and have to tap the same pool of firms for money over time. And the firms need to be perceived as founder friendly in order to cozy up to the most promising deals.

Mr. Kalanick’s two new appointees are well known in the business world. As chief executive of Xerox, Ms. Burns was the first African-American woman to run a Fortune 500 company. Ms. Burns, 59, received a master’s degree in mechanical engineering from Columbia University and worked at Xerox her entire career, beginning as an intern in 1980 and becoming the head of the company in 2009.

Mr. Thain, 62, was one of Wall Street’s best-known figures until the financial crisis hit Wall Street in 2008. He became the head of the New York Stock Exchange in 2004, then the chief executive of Merrill Lynch in 2007. He sold the firm to Bank of Americanduring the financial crisis and was later chief executive of CIT Group, a lender to small and midsize businesses, until he retired in 2015.