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 latest tweet from Musk came the same day that US District Court judge Alison Nathan from NY requested a joint letter regarding why she should approve of the tentative settlement between Musk the Securities and Exchange Commission.
"Musk's false and misleading public statements and omissions caused significant confusion and disruption in the market for Tesla's stock and resulting harm to investors", the SEC said in its complaint.
Following the SEC settlement, Musk agreed to pay a $20m fine and step aside as Tesla's chairman for three years.
Reuters couldn't reach out to Murdoch for comment.
Tesla did not immediately respond to a request for comment on what Musk meant by the tweet.
FILE PHOTO: Tesla Chief Executive Elon Musk stands on the podium as he attends a forum on startups in Hong Kong, China January 26, 2016. The company 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.
After days of restraint, Trump mocks Kavanaugh accuser
McCaskill announced she will be a "no" when the full Senate votes on Kavanaugh's confirmation, the National Review reported . President Trump, as well as his family, have raised fears about the possibility of politically motivated false accusations.
BlackRock declined to comment on Musk's tweet.
The SEC alleged that Musk said a representative from Saudi Arabia's Public Investment Fund had shown interest in taking Tesla private but had never discussed with Musk any of the specific terms he described on Twitter, including a proposed $420 share price and an option for all existing Tesla shareholders to remain with the company after it went private. The SEC had accused him of securities fraud in a lawsuit and had asked a federal judge to bar him from ever again leading a publicly traded company.
"Keep it coming, Elon", one person said.
He has used Twitter to criticize short-sellers betting against his company.
Later Thursday, Musk returned to Twitter to deride short-sellers.
For her part, Nathan may have limited room to intervene, after a federal appeals court curbed the ability of judges to reject SEC settlements.
Greenlight is having a hard year, with its main fund down 25.7 percent through September, but said its Tesla short position was its second biggest victor in the third quarter.
The SEC settlement still needs court approval. The court expects to receive the joint statement by 11 October, explaining why their settlement should be approved.
The case is SEC v Musk, U.S. District Court, Southern District of New York, No. 18-08865.