Twitter's shares rose by about 4 percent in opening trade in New York on speculation the firm could agree a takeover deal with Elon Musk as early as Monday.
Elon Musk at the TED2022: A New Era conference in Vancouver, Canada on 14 April. Photo: AFP / TED Conferences / Ryan Lash
The board of the social media giant is reported to have met with Musk over the weekend to discuss his US$43 billion offer.
Twitter had initially rebuffed the billionaire's approach.
But according to reports, Twitter is now in the final stages of negotiations with Musk to buy the business.
News agency Reuters 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 percent to US$50.62 on the reports, but remained below Musk's offer price of US$54.20 a share.
Musk's targeting of Twitter has moved at remarkable speed. It emerged at the beginning of April that Musk, who is the boss of electric carmaker Tesla, had become the largest shareholder in Twitter with a 9.2 percent 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 Musk. The mechanism would have created difficulties for Musk if he attempted to increase his stake in the company.
The board now appears to have U-turned after Musk revealed he had lined up a US$46.5b 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.
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 percent of those who responded wanted an edit button.
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!".
He also recently tweeted: "If our Twitter bid succeeds, we will defeat the spam bots or die trying", "And authenticate all real humans."
It is not clear who would run Twitter if Musk's bid succeeds. He is the chief executive as well as the "Technoking" of Tesla according to the company's website.
He is also chief executive, chief technology officer and chairman of rocket and spacecraft firm SpaceX. Musk is also involved in a number of other businesses and start-ups.
Twitter is led by chief executive Parag Agrawal, who took on the role last November with the backing of co-founder and former boss Jack Dorsey.
At the time, Dorsey said: "My trust in Parag as Twitter's chief executive is deep. I'm deeply grateful for his skill, heart, and soul. It's his time to lead."
Musk has been critical of Twitter's management. In his offer document, he told Twitter's chairman Bret Taylor: "I don't have confidence in management."
Agrawal has not discussed the takeover on the platform since earlier this month, when he announced that Mr Musk had turned down an offer to sit on the company's board, a decision he said would be "for the best".
- BBC