Elon Musk could face civil and criminal penalties if found he didn't secure financing at the time of his tweet about taking Tesla private, former Securities and Exchange Commission Chairman Harvey Pitt told CNBC on Wednesday.

"If you make a false statement in connection with the trading of securities, you run the risk of both having to pay for the damages you caused and also you run the risk of a criminal prosecution," Pitt, now CEO of consulting firm Kalorama Partners, told "Squawk Box."

Tesla did not immediately respond to CNBC's request for comment.

Shares of the electric car maker soared Tuesday after Musk's tweet about possible plans to take the company private, as well as reports of a $2 billion investment in the automaker from a Saudi Arabia investment fund.

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In a company blog post later Tuesday, Musk made a more formal statement about privatizing Tesla.

The tweets stunned investors, with some questioning whether the CEO could face lawsuits if it were proven he didn't secure financing.

Pitt said the SEC could compel Tesla employees to make a formal statement. Tesla shouldn't wait for a call like that, Pitt said, adding the company should be "supplying information to the SEC before they even ask that."

"There's enough here for people to ask questions," Pitt said. "Whether or not he's done anything illegal is far from certain."

Taking Tesla private is a pretty good strategy, even if how the CEO announced it wasn't, Jed Dorsheimer, senior analyst at Canaccord Genuity, told CNBC.

"If we take the emotions out of the unconventional nature of how he went about announcing it, [we can] recognize that this is pretty common for him to announce things on Twitter," Dorsheimer told "Squawk Box."

Dorsheimer, who said he can't speak to the legality of Musk's tweets, said his firm previously put a hold on Tesla due to Musk's "unpredictability." But Dorsheimer reiterated that privatization could help the company.

Going private could be a way for Tesla to avoid close scrutiny by the public market.