Vote on his future as its chief executive.
In a poll to his 122 million followers, he tweeted: “Should I step down as head… I will abide by the results.”
The technology tycoon, who also runs Tesla and Space X, has faced much criticism since taking over Twitter.
After a legal battle, Mr Musk took control of the company in October in a $44bn ($36bn) deal.
At 07:00 GMT on Monday, more than 13 million people had voted in his poll and 57% had said yes to Mr Musk stepping down.
Dan Ives, senior equity analyst at Wedbush Securities, told the BBC he believed the vote would “ultimately” lead to the “ending of Musk’s reign as chief executive of Twitter”.
There has been a flurry of controversial changes at Twitter since Mr. Musk bought the social media site in October.