Talks at OpenAI to bring back Sam Altman, the artificial intelligence start-up’s recently ousted chief executive, continued on Sunday afternoon but there were disagreements over the makeup of the company’s board of directors, according to two people familiar with the discussions.

Mr. Altman, 38, spent the weekend waging a pressure campaign on the start-up’s four-person board of directors who ousted him on Friday afternoon, three people familiar with the matter said. The result was a groundswell of support from investors, employees and OpenAI executives.

Mr. Altman was at the OpenAI headquarters on Sunday afternoon. He posted a photo of himself on X, formerly Twitter, wearing a guest identification badge and said, “first and last time am ever wear one of these.”

The negotiations included a look at how the company’s board of directors might be reshaped if Mr. Altman returns as chief executive, two of the people said. Members of the board have not yet agreed to what a restructured board of directors might look like — nor is Mr. Altman’s reinstatement an inevitability, two of the people said.

There has been a whirlwind of activity since Mr. Altman was forced out of OpenAI, a company he helped found eight years ago that has become one of the most closely watched companies in tech thanks to its popular ChatGPT chatbot.

On Sunday afternoon, Will Hurd, a former OpenAI board member and a former Republican congressman from Texas, was standing outside the company’s headquarters in San Francisco’s Mission district waiting for a ride to the airport after spending two days digging into the details of Mr. Altman’s dismissal.

Mr. Hurd said that a representative of the company had called him on Friday morning, before Mr. Altman’s removal, and asked for his help in navigating the leadership upheaval. Mr. Hurd traveled from Texas to San Francisco on Saturday.

“The industry is important, the company is important,” Mr. Hurd said. “This is the future. How do we make sure there’s a level of trust and transparency? All things we want from models, we want from governance.”

OpenAI declined to comment.

OpenAI’s board of directors is unique. The organization started as a nonprofit before transforming itself into a for-profit company and bringing on Microsoft as its biggest investor. The for-profit company still answers to the nonprofit board. As a result, the company’s investors have had no official say in what happens to the start-up or who leads it.

Before Mr. Altman was forced out, OpenAI had six board members, including Mr. Altman and Greg Brockman, a company co-founder and board chairman who quit on Friday in solidarity with Mr. Altman.

The other board members are Ilya Sutskever, an OpenAI co-founder; Adam D’Angelo, the chief executive of Quora, the question-and-answer site; Helen Toner, a director of strategy at Georgetown’s Center for Security and Emerging Technology; and Tasha McCauley, an entrepreneur and computer scientist.

Since Friday, people close to the company have been trying to learn why the board dismissed Mr. Altman. Brad Lightcap, the company’s chief operating officer, said in a note to staff on Saturday that there was no “malfeasance” involved.

The stalemate is the latest twist in a series of power struggles at OpenAI. The fight drew attention to a longtime rift in the A.I. community between people who believe A.I. is the biggest business opportunity in a generation and others who worry that moving too fast could be dangerous.

The company was recently in talks to raise a new funding round that would value the company at more than $80 billion. Bloomberg News earlier reported some of the details of the discussions.

Mr. Altman’s potential reinstatement would be a dramatic reversal for OpenAI, which said he had not been “consistently candid” in his discussions with the board when it unceremoniously ousted him.

As deliberations continued on Sunday, executives at OpenAI called in resources. At 12:45 p.m., a deliveryman with a dozen drinks from the Boba Guys chain showed up on a motorbike outside with two bags. Another deliveryman followed later with a half dozen more drinks.

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