The board of the social media giant is reported to have met with Mr Musk over the weekend to discuss his $43bn offer.
Twitter had initially rebuffed the billionaire’s approach.
But according to reports, Twitter is now in the final stages of negotiations with Mr Musk to buy the business.
Reuters, the news agency, reported that an announcement may come later on Monday after Twitter’s board met and recommended a deal to its shareholders.
Shares rose almost 4% to $50.62 on the reports, but remained below Mr Musk’s offer price of $54.20 a share.
Mr Musk’s targeting of Twitter has moved at remarkable speed. It emerged at the beginning of April that Mr Musk, who is the boss of electric carmaker Tesla, had become the largest shareholder in Twitter with a 9.2% stake.
He was then invited to join Twitter’s board but turned down the offer before launching a surprise bid for the company on 14 April, saying he wanted to “unlock” the social media platform’s “extraordinary potential”.
Twitter responded by putting in place a so-called “poison pill” defence to fend off Mr Musk. The mechanism would have created difficulties for Mr Musk if he attempted to increase his stake in the company.
The board now appears to have U-turned after Mr Musk revealed he had lined up a $46.5bn financing package for his offer – with funding from a mixture of his own assets and the backing of Wall Street banking giant Morgan Stanley and other firms.
Mr Musk recently said that he believes that Twitter should be more open and transparent. “I think it’s very important for there to be an inclusive arena for free speech,” he told the TED2022 conference in Vancouver.
The billionaire, who is the world’s richest man according to Forbes magazine, has also hinted at his plans for Twitter.
He asked his 83.3 million followers on Twitter if the social media platform should allow people to edit their posts – 74% of those who responded wanted an edit button.
Mr Musk’s has also criticised Twitter’s process of verifying a Twitter account which he said was like “if Kafka had a Magic 8-Ball!”.
