Murdoch, the outgoing 21st Century Fox chief executive and a Tesla board member, was the lead candidate to replace Musk as Tesla chairman, the FT reported on Wednesday, citing two people briefed on the discussions.
Mr Murdoch is now chief executive of the United States media giant 21st Century Fox, but he will step down after the firm completes the sale of much of its business to Walt Disney.
Neither company immediately responded to Deadline's request for comment.
Mr Musk agreed to give up the chairmanship last month to resolve claims of fraud brought by United States financial regulators. He's still CEO, however. The regulator sued Mr Musk last month over allegations that he misled the stock market and harmed investors by tweeting in August that he had "funding secured" to take Tesla private. Vehicles have piled up in lots around California awaiting transport, and Musk said last month that Tesla had moved from "production hell to delivery logistics hell". Atherton Research says Elon Musk's auto company sold 69,925 of its Model 3, S and X cars, while Mercedes Benz sold 66,542 vehicles (excluding commercial vans) in the U.S.
Earlier this year, Elon Musk created the idea of a Tesla-branded tequila, cleverly named Teslaquila, as part of an April Fools' Day joke. He also stepped down as chairman of Sky after its takeover by Comcast.
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Murdoch will not join his father, Rupert Murdoch, and his brother Lachlan at "New Fox" - the entity remaining after 21CF's sale of assets to Disney is completed.
Since Tesla has until the end of the year to hand out full tax credits, a further rush of orders could drag down the company's already strained production and delivery chain.
Tesla faces a crucial moment as it struggles to produce its Model 3 sedan in large volume and deliver it quickly to customers amid an ongoing cash crunch that has concerned some analysts.
Under terms of the settlement, Musk is not permitted to be re-elected to the chairman post for three years.
Tesla shares were down 1.3 percent at $253.5 in premarket trading.