Some Tesla directors have proposed that James Murdoch, fellow board member at the electric carmaker and chief executive officer of Twenty-First Century Fox Inc., succeed Elon Musk as its chairman, the New York Times reported, citing people involved in the board's discussions.

Murdoch hasn't volunteered for the post nor has he discussed it with any other director, the newspaper reported.

The board has not engaged in any "serious" discussions of who should be the chairman, the paper said, citing a source close to the selection process.

The New York Times also said Musk had threatened to resign, in a phone call with directors, if the board insisted that he and the company enter into a settlement with the Securities and Exchange Commission (SEC).

However, the next day, Tesla's attorneys were back at the SEC requesting them for a second chance, this time with Musk's 'grudging approval', the paper said.

Tesla (TSLA) and Musk reached a settlement with regulators on Saturday, where they agreed to pay $20 million each to financial regulators and the billionaire will step down as chairman but remain as CEO.

Reuters couldn't reach out to Murdoch for comment. Tesla and Twenty-First Century Fox were not immediately available for comment outside regular business hours.