The U.S. Securities and Exchange Commission has made inquiries to Tesla over CEO Elon Musk's tweets about taking the company private, reported Dow Jones on Wednesday, citing sources.

On Tuesday, Musk said on Twitter that he was considering taking Tesla private at $420 a share, and that he had already secured the funding.

But since then he has provided no details on the source of the funding, which would be substantial. At $420 a share Tesla would be valued at about $71 billion.

UBS analyst Colin Langan estimated a take-out price of $88 billion, which includes debt, stock-based awards and warrants, among other things. That would make it twice as large as the previous largest leveraged buyout, TXU, which was valued at $44.37 billion, including debt.

Following the series of tweets, Tesla posted a letter Musk had sent to employees that said he wants to take Tesla private to remove the distracting attention Tesla receives from the public, including short sellers. Tesla has generated a great deal of interest and enthusiasm for its electric cars and other products, but the company has burned through billions, rarely made a profit, and has repeatedly missed production targets.

The potential downside of going private is a lack of access to capital markets, and many on Wall Street expect Tesla will need more capital to fund its ambitious growth plans, which include building factories around the world, continuing to ramp up production of its current vehicles, launching new vehicles, and building a business in solar power and energy storage.

The SEC asked whether the statements Musk made on Tuesday were true and why he chose to disclose his plans on Twitter, Dow Jones reported, citing sources.

Tesla and the SEC did not immediately respond to a request for comment from CNBC.

Musk is not prohibited from disclosing his plans on social media, however the move was widely seen as being highly unusual, especially since Musk already announced the price at which he wants to buy out shareholders.

"That is what is so weird about this" Wayne University law professor Peter Henning told CNBC. "It's backwards corporate governance. You don't announce a price and then negotiate."

Legal experts told CNBC the main legal question is whether what Musk said about his plans was accurate, and whether he left out any information he should have disclosed.

That especially applies to Musk's claim that he had already secured the needed funding.

"He is not prohibited from making that statement as long as it is truthful and not manipulative" said Tom Gorman, a partner at law firm Dorsey and Whitney. "You are talking about billions of dollars of funding. He said he has it. If he has it, he has it. If he doesn't that is a problem, because then he would be making at minimum a false statement. That could have an obvious impact he would certainly be aware of."

Tesla shares closed down more than 2 percent at $370.34.

Read the full story at the Wall Street Journal.

Correction: At $420 a share Tesla would be valued at about $71 billion. An earlier version misstated what Tesla's valuation would be.