The agreement also called for two new independent directors to join Tesla's board.
It is reported that Murdoch hasn't volunteered for the post nor has he discussed it with any other director.
"Just want to that the Shortseller Enrichment Commission is doing incredible work".
Last week Musk settled the securities fraud charge from the SEC, following his tweet in August about potentially taking Tesla private.
After initially refusing an offer to settle the case, Mr Musk backed down, agreeing that he and his company would each pay a $20m fine, he would step down as Tesla's chairman (but remain chief executive), and that he would allow oversight of his communications about company news.
As a result, the SEC reportedly increased Musk's fine from $10 (£7.7 / €8.7) million to $20 (£15.4 / €17.4) million.
"Just want to [sic] that the Shortseller Enrichment Commission is doing incredible work", Musk, a frequent critic of investors betting against the electric auto company, wrote on Twitter.
Tesla declined to comment.
Tesla and the SEC declined requests for comment. The company's board reportedly hasn't had any serious discussions on who will replace Musk, but some board members have thrown their support behind James Murdoch who is the son of Rupert Murdoch and the CEO of 21st Century Fox.
Tesla shares dropped 4.4 percent Thursday, and were down an additional 2.5 percent in extended trading, to $274.75.
"In and of itself it's not an ominous sign", said Jordan Thomas, a partner at Labaton Sucharow and former SEC lawyer.
One of the terms of the settlement was that Musk would not take any action or make any public statement denying any SEC allegations or "creating the impression that the complaint is without factual basis".
But Musk and Tesla might not even be out of the woods yet. The proposed take-private price was $420 a share.
The BBC's North America technology reporter Dave Lee said many people had expected Mr Musk would "rein in his Twitter habit".