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.
Musk's tweet came the same day that Judge Alison Nathan, a US District Court judge in NY, requested a joint letter explaining why she should approve the tentative settlement deal between Musk and the Securities and Exchange Commission. The SEC did not respond to requests for comment on Nathan's order and Musk's tweet.
Reuters couldn't reach out to Murdoch for comment. There was no immediate reaction from the SEC.
The SEC alleged that Musk hadn't locked up the estimated $25 billion to $50 billion that it would have required to pull off that deal, and wanted to punish him by forcing him out as Tesla's CEO.
Tesla also accepted a $20 million fine, despite not being charged with fraud.
It's not an uncommon practice for a judge to ask for such a letter.
The SEC declined to comment Thursday.
Shares of Tesla closed down $12.97, or 4.4 percent, at $281.83, and fell another 2.1 percent to $276 following Musk's tweet after market hours.
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Following the investigation into Fan , authorities have declared a wider crackdown on the entertainment industry, Xinhua said. She had disappeared from public view in June following reports she was being investigated for tax fraud .
"They promised to have controls to keep him from doing these things", she said.
Musk and the SEC came to an agreement late last week to settle charges alleging that the billionaire had defrauded investors in statements relating to an August 7 tweet about bringing the company private with "funding secured".
He has used Twitter to criticize short-sellers betting against his company.
Musk continued his tirade against short-sellers by sending out another inflammatory tweet at 6:40 p.m.
Some judges have complained about being viewed as rubber stamps for SEC settlements. He called Musk at the request from one of Musk's lawyers, the Journal reported.
But in 2014, the 2nd U.S. Circuit Court of Appeals overturned Rakoff's rejection of a $285 million SEC settlement with Citigroup Inc, saying he should have given "significant deference" to the regulator.
That arrangement is unusual, says Jay Dubow, a partner at Pepper Hamilton and a former branch chief with the SEC's enforcement division.
The case is SEC v Musk, U.S. District Court, Southern District of New York, No. 18-08865.
Judge Alison Nathan gave both sides until October 11 to give their explanation through a joint statement.