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.
These included a tweet saying there was "funding secured" to take the Palo Alto, California-based company private at $420 per share.
The settlements were related to Musk's infamous "funding secured" tweets regarding a potential move to take Tesla public.
Tesla and Mr. Musk agreed to pay $20-M each to the regulator under the settlement, and Mr. Musk to step down as the company's Chairman but remain CEO as part of the deal.
Reuters couldn't reach out to Murdoch for comment.
Elon Musk is again under the limelight as the Securities and Exchange Commission has accused him of the fraud and has filed a lawsuit. Separately, Tesla agreed to pay $20 million to settle claims it failed to adequately police Musk's tweets.
The judge said it was her regular practice to request such letters.
Tesla declined to comment.
The tweet appeared to jar investors, who drove Tesla shares down 2.4 percent in after-hours trading to $275.
"She may want to know why Tesla is paying a fine because the CEO doesn't know when to shut up", said Adam Pritchard, a University of MI law professor and former SEC lawyer.
Former SEC lawyers questioned the wisdom of Musk's latest tweet, but said it was unlikely to jeopardize the settlement, which prevents Musk from denying wrongdoing or suggesting that the regulator's allegations were untrue. "Maybe a 420M fine would have been more appropriate?" one Twitter follower said.
Musk continued his tirade against short-sellers by sending out another inflammatory tweet at 6:40 p.m.
Exactly. Short-sellers are value destroyers.
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.
Some judges have complained about being viewed as rubber stamps for SEC settlements.
The SEC settlement still needs court approval. "The vast majority of settlements like this are approved by courts". "This is just a hoop to be jumped through".