The extraordinary claims, first reported by the Financial Times, have been denied by Mr Musk.
Just hours after the story posted, Mr Musk replied to the FT’s story on Twitter saying: “This is incorrect.”
Tesla has until November 13 to appoint a new chairman of the board as per the agreements made between the company and the US Securities and Exchange Commission.
The SEC said that when Mr Musk had tweeted about taking the company private he had committed fraud.
The agreement reached between Tesla and the SEC states that while Mr Musk can remain as the CEO of the company, he must give up his chairmanship.
The SEC investigated after the CEO had tweeted he was planning on taking his company private at $420 a share, with the price being a reference to marijuana.
According to the FT, Mr Murdoch is already an independent director of Tesla and he has expressed interest in the chairmanship.
The Fox mogul Rupert Murdoch’s son joined the board last year after previously working with various media companies.
The younger Murdoch has no experience when it comes to manufacturing and has yet to lead a company that makes any type of vehicle.
While Mr Musk is required to relinquish control for several years, there are concerns that no one will be able to match his personality.
CEO of the corporate governance consulting company Boardspan, Abby Adlerman said: “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?’”
Earlier this year, the leading US proxy advisors Glass Lewis & Co and International Shareholder Services, along with the union-affiliated investment advisor CtW Investment Group, all recommended that investors vote “against” Mr Murdoch’s re-election as a Tesla director during the companies annual June 5 meeting.
CtW said that this was due to his lack of experience and his “troubled history as an executive and director,” both of the proxy firms said that he served on too many boards.
While Mr Murdoch stepped down from Sky Plc’s board after Comcast Corp’s takeover, he still sits on the boards for Twenty-First Century Fox and News Corp.
In 2012 he was forced out as chairman of Sky, previously BSkyB, after the infamous phone-hacking scandal.
He returned in 2016 after working for several years at Fox.
Courteney Keating, Glass Lewis’ research director, expressed concern that even if Mr Murdoch left Sky, he would still not be all to meet the time commitments required.
She said: “I would still have reservations about the time he would be able to allot to this”
The executive director of CtW Investment Group Dieter Waizenegger argues that it is important to consider multiple candidates because “investors should really be wary about the board making the easy choice here” in opting for Mr Murdoch.