Elon Musk is offering to buy Twitter, just days after the Tesla CEO said he would no longer be joining the social media company’s board of directors.

Twitter said in a regulatory filing on Thursday that Mr Musk, who owns slightly more than 9 per cent of its stock and is the company’s biggest shareholder, provided a letter to the company on Wednesday that contained a proposal to buy the remaining shares of Twitter that he doesn’t already own.

Musk offered $US54.20 per share of Twitter’s stock, which equates to $US41 billion ($55 billion).

He called that price his best and final offer, although the billionaire provided no details on financing.

“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,” Mr Musk says in the filing.

“However, since making my investment I now realise the company will neither thrive nor serve this societal imperative in its current form. Twitter needs to be transformed as a private company.”

The buyout offer from Musk is just the latest development in his relationship with Twitter.

The billionaire revealed in regulatory filings over recent weeks that he’d been buying shares in almost daily batches starting January 31.

Only Vanguard Group’s suite of mutual funds and ETFs controls more Twitter shares.

Mr Musk has been a vocal critic of of Twitter in recent weeks, mostly over his belief that it falls short on free speech principles.

The social media platform has angered followers of Donald Trump and other far-right political figures who have had their accounts suspended for violating its content standards on violence, hate or harmful misinformation.

Mr Musk also has a history of his own tweets causing legal problems.

Mr Musk said last week that he informed Twitter he wouldn’t be joining its board of directors five days after he was invited.

He didn’t explain why, but the decision coincided with a barrage of now-deleted tweets from Musk proposing major changes to the company, such as dropping ads – its chief source of revenue – and transforming its San Francisco headquarters into a homeless shelter.

Mr Musk left a few clues on Twitter about his thinking, such as by “liking” a tweet that summarised the events as Musk going from “largest shareholder for Free Speech” to being “told to play nice and not speak freely”.

After Mr Musk announced his stake, Twitter quickly gave Musk a seat on its board on the condition that he not own more than 14.9 per cent of the company’s outstanding stock, according to a filing. But Mr Musk backed out of the deal.

Mr Musk’s 81 million Twitter followers make him one of the most popular figures on the platform, rivalling pop stars like Ariana Grande and Lady Gaga.

But his prolific tweeting has sometimes gotten him into trouble with the US Securities and Exchange Commission and others.

Mr Musk and Tesla in 2018 agreed to pay $US40 million ($53 million) in civil fines and for Mr Musk to have his tweets approved by a corporate lawyer after he tweeted about having the money to take Tesla private at $US420 per share.

That didn’t happen but the tweet caused Tesla’s stock price to jump.

Mr Musk’s latest trouble with the SEC could be his delay in notifying regulators of his growing stake in Twitter.

Shares of Twitter jumped 11 per cent before the market opened.

The stock is still down from its 52-week high of about $US73.