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.

Image Ursula Burns, a former chief executive of Xerox, is one of Mr. Kalanick’s new appointees to the Uber board of directors. Credit... Scott Olson/Getty Images

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.