The Fox mogul Rupert Murdoch's son joined the board past year after previously working with various media companies.
The current favorite, according to a report on Wednesday by the Financial Times, is James Murdoch, chief executive of 21st Century Fox and a current member of the Tesla board. The SEC intends to distribute the penalties to affected investors, it said.
Tesla's board has been criticized for lacking independence, and while Murdoch has frequently sided with Musk in matters before the board, he remains financially independent in a way that the other major candidate for the job, Antonio Gracias, isn't.
Musk must relinquish his role as chairperson under a settlement with the Securities and Exchange Commission (SEC) over his tweeting, which left open the possibility that the carmaker could appoint an existing independent director.
The settlement allows Musk to remain as CEO of Tesla.
Tesla didn't respond to requests for comment, but Musk disputed the report's accuracy in a tweet Wednesday afternoon, saying simply: "This is incorrect". The deal with the SEC eased uncertainty over Tesla's future because the agency's lawsuit initially sought to bar Musk from serving as an officer or director of a public company, a prospect that rattled investors.
The question when it comes to James Murdoch is, 'Is he the guy who'll be able to establish that level of authority with Elon Musk?'
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He joined Tesla's board previous year as a non-executive director and has reportedly said he wants the job of chair.
Two proxy advisors, Glass Lewis & Co and Institutional Shareholder Services, advised investors to vote against reelecting Murdoch as a director in June, citing his lack of experience in the field, his "troubled history as an executive and director", and his presence on "too many boards". He is now on the boards of Twenty-First Century Fox and News Corp. He stepped down from the board of Sky Plc on Tuesday following the completion of Comcast Corp's takeover of the broadcaster.
Murdoch was forced out as chairman of Sky, then BSkyB, in 2012 after being embroiled in Britain's phone-hacking scandal. The terms of that agreement included Tesla appointing a new independent chairman by the middle of November.
He returned in 2016 after working for several years at Fox.
Glass Lewis research director Courteney Keatinge said on Wednesday that while Murdoch's departure from Sky could alleviate some concerns, the Tesla chairmanship would still require a big time commitment as the company faces pressures on many fronts. Musk remains CEO of Tesla.
James Murdoch is now chief executive of 21st Century Fox, but will leave the role when the entertainment giant completes the sale of the majority of its assets to Disney, and will be succeeded by his brother Lachlan.