Musk offered $54.20 a share, which values the social media firm at $43.4 billion, in a filing dated Wednesday April 13 with the Securities and Exchange Commission.
Twitter's board said it would carefully review what it termed Musk's "unsolicited, non-binding" offer and decide on a course of action that was "in the best interest of the Company and all Twitter stockholders."
Musk's latest move toward Twitter comes just days after he turned down a seat on the board following his acquisition of a 9.2 percent stake in the microblogging platform.
"I invested in Twitter as I believe in its potential to be the platform for free speech around the globe, and I believe free speech is a societal imperative for a functioning democracy," Musk said in his filing.
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"However, since making my investment I now realize the company will neither thrive nor serve this societal imperative in its current form," he said.
"Twitter needs to be transformed as a private company. As a result, I am offering to buy 100% of Twitter for $54.20 per share in cash, a 54% premium over the day before I began investing in Twitter."
'Popcorn time' Musk, Twitter's biggest shareholder, said his "offer is my best and final offer" and he would reconsider his position as a shareholder if it was rejected.