The Securities and Exchange Commission has opened a probe to determine whether Musk's comments were based in fact, the Wall Street Journal reported.
Tesla didn't respond to requests for comment on Wednesday.
SoftBank has previously invested in Uber, Nvidia and GM's Cruise.
After Musk made the tweet, trading of Tesla shares was temporarily halted as an official announcement was expected to follow shortly thereafter.
Some Wall Street analysts were skeptical of Musk's ability to gather the huge financial backing to complete such a deal, given that Tesla loses money, has $10.9 billion of debt and its bonds are rated junk by credit ratings agencies.
At $420 apiece, buying all of Tesla's shares would cost about $72 billion, but Morgan Stanley analyst Adam Jonas wrote in a note to investors Wednesday that he expects about $50 billion in additional net debt. "And if you stay as a shareholder you get less information than before and you depend more and more on Elon Musk". The board said they ended up meeting several times to discuss the issue and are now "taking the appropriate next steps to evaluate" the proposal.
Six of nine members said in a prepared statement Wednesday that Musk began talking with the board about the move last week.
Tesla is now valued at $63.8 billion.
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"If Apple wanted to be in the vehicle business, this is how you do it-you step up, doesn't have to be $420 (per share), it could be another price".
Ultimately it will be up to the shareholders whether Tesla remains public or goes private, but the SEC is now curious about how Tuesday's events unfolded.
The deal would be the biggest leveraged buyout of all time, beating the $45 billion record set by Texas power utility Energy Future Holdings. Musk tweeted that he was considering taking the electric auto giant private at $420, which implies a 12% premium from where TSLA is trading.
Mr Musk and SoftBank held unsuccessful talks about a take-private deal in April 2017, according to a source familiar with the matter.
He's also Tesla's largest shareholder.
The news comes one day after Musk said he was considering taking Tesla private.
Just last week, he revealed he had been working 110 hours a week to deliver on short-term promises he had made to Wall Street, a load he traced to his boorish behaviour toward two analysts earlier this year.
Scepticism about the legitimacy of Musk's proposed deal surfaced nearly immediately after Musk dropped the bombshell on his Twitter account a few hours after the stock market opened on Tuesday.
Those who believe Musk is carrying out a vendetta against short sellers may point to a May 4 tweet suggesting he might have something up his sleeve.