"Am considering taking Tesla private at $420". If Musk were to make good on his tweet, it is unclear from where he could secure funding to buy out Tesla's shareholders.
Tesla's shares were up 6.5pc at $363.46.
- CBS News' Jillian Harding contributed reporting.
A deal at $420 per share represents a 22.8 percent premium to Tesla's closing price on Monday, valuing the company at about $72 billion.
While Mr Musk has previously discussed the drawbacks of being a public company, he has also used his Twitter account in ways that surprised investors.
Tesla did not immediately respond to a request for comment. Shares are up as high as 9% on the day.
On April Fool's Day, Mr Musk, who owns nearly 20% of the company, joked on Twitter about Tesla going bankrupt.
Analysts took Musk at his word. I think he's serious. The stock has fallen over the past year as losses mounted.
Short interest in Tesla on Monday stood at almost $12 billion, equivalent to 28 percent of its float, according to S3 Partners, a financial analytics firm.
The tweets came after a report in the Financial Times that Saudi Arabia's sovereign wealth fund had taken a 3%-5% stake in Tesla, a holding worth at least $1.9bn.
"Details, structure, participants and how the valuation has been determined remain to be seen".