Some Wall Street analysts are skeptical of Musk's ability to gather the financial backing to complete such a deal.
The SEC declined to comment on the reports.
Mr Musk has been under intense pressure this year to turn his money-losing, debt-laden company into a profitable higher-volume manufacturer, a prospect that has sent Tesla's valuation higher than that of GM. However, Musk expects the shareholders to agree with him and allow him to take the company private again.
The Wall Street Journal reported Wednesday that SEC investigators could be looking into a variety of violations such as artificially raising the automaker's value via the Tweet and failure to disclose the source of funding to shareholders.
The tweet was only 61 characters long, but it was powerful enough to make Tesla CEO Elon Musk $1.4 billion richer. Trading in Tesla's stock was eventually halted until everyone could figure out what was going on.
Like Musk's tweet and email to employees, the directors' statement didn't contain details on how the buyout would be financed.
Musk is already CEO of another private company: SpaceX.
Lawyers said every word of Musk's tweets will be examined, including why he used words like "considering", to determine whether he might have tried to inflate the stock price to blow out short-selling investors, a longtime foe. Who will fund a buyout? Company share prices, particularly in deals as giant as this one, are nearly always decided by corporate executives, board members or consultants who review market data and optimize for maximum value.
Shares rose 11 percent Tuesday but fell almost 2 percent at midday Wednesday to $372.38.
Musk has feuded publicly with regulators, critics, short sellers and reporters, and some analysts suggested that less transparency would be welcomed by Musk. "Oh and uh short burn of the century comin (sic) soon, " he wrote.More news: Liverpool complete impressive pre-season with Torino win
An SEC investigation would be the latest among a string of recent hiccups for Musk and Tesla.
Some of the fervour for a Tesla buyout faded on Wednesday.
The stock fell more than 2% and closed at $370.34. "Finally, as the most shorted stock in the history of the stock market, being public means that there are large numbers of people who have the incentive to attack the company".
Musk has promised to begin making money on a consistent basis starting in the third quarter, escalating the pressure that he has already publicly acknowledged has taken an enormous toll on him.
Some on Wall Street shared that view.
The billionaire has at least been clear about his resentment of life on the public market.
Buying Tesla in its entirety would cost US$72b, based on the company's outstanding stock as of July 27.
Musk also said he intends to give Tesla's existing shareholders the option of retaining a stake in the company through a special fund if they want to stay invested.
"Shareholders could either to sell at 420 or hold shares & go private", he later added.
Net net, Rusch reiterates an Outperform rating on Tesla shares, with a price target of $385, which represents a slight upside potential from current levels.
In a Wednesday research note, Morgan Stanley analyst Adam Jonas estimated Tesla will end up taking on about US$50b in additional debt if the company goes private.