The transaction would amount to the largest leveraged buyout - in which a company borrows money to repurchase shares and exit the public market - of all time.
Musk has 22.3 million followers on Twitter, and Tesla's share price was already rising after the Financial Times reported that Saudi Arabia's sovereign wealth fund had taken a $2bn stake in the company.
Tesla shares fell 2.4 percent to $370.34 on Wednesday after closing up 11 percent on Tuesday.
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.
It's unclear if Musk has misled investors, let alone whether he has done so intentionally.
Tesla CEO Elon Musk tweeted Tuesday that he's is mulling taking Tesla private at US$420 a share. He now owns 20 percent of the company, so more than $60 billion is needed to buy the business from public shareholders.
Former SEC chair Harvey Pitt told CNBC that while the U.S. stock market regulator permits executives of publicly listed companies to use social media to make statements about their businesses, Musk's tweet was still "highly unprecedented".More news: Manafort defense questions star witness Gates about 'secret life'
"I'm trying to accomplish an outcome where Tesla can operate at its best, free from as much distraction and short-term thinking as possible, and where there is as little change for all of our investors, including all of our employees, as possible".
Musk's proposed move is seen as a strategy to avoid several issues: Wild stock swings, short-sellers who profit on stock value drops and shareholder pressure on quarterly performance, according to the Financial Times.
The key question following Musk's Tuesday tweet-storm was whether current investors, as represented by the board of directors, were interested in such a proposal.
China's Tencent Holdings Ltd, which took a 5-percent stake in Tesla past year, could also be a possible partner.
Elon Musk, founder, CEO and lead designer at SpaceX and co-founder of Tesla, speaks at the International Space Station Research and Development Conference in Washington, U.S., July 19, 2017.
"This proposal to go private would ultimately be finalized through a vote of our shareholders", Musk said in explaining the potential move. Tesla spokesmen didn't immediately respond to requests for comment on Wednesday. "And if you stay as a shareholder you get less information than before and you depend more and more on Elon Musk".
Days after Tesla opened up the Model 3 configurator to all potential buyers, not just reservation holders who got a place in line over a year ago, uncomfortable math lingers over the automaker's.