Some Tesla directors have proposed that James Murdoch, fellow board member at the electric carmaker and chief executive officer of Twenty-First Century Fox, succeed Elon Musk as its chairman.
This is according to the New York Times, which cited people involved in the board's discussions.
James Murdoch has not 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 and Musk reached a settlement with regulators on Saturday, where they agreed to pay $20m each to financial regulators and the billionaire will step down as chairman but remain as CEO.