OCT 5: In a letter to the firm, Mr Musk agreed to pay the price he offered months ago before trying to quit the deal.
The surprise reversal comes just weeks before the two sides were due in court.
Twitter, which had sued Mr Musk to force the takeover to move forward, was seen as having the stronger case.
In the letter, attorneys for Mr Musk said he intended to move ahead to complete the transaction, pending receipt of the financing and an end of the legal fight.
A spokesperson for Twitter acknowledged the firm had received the proposal, adding "the intention of the company is to close the transaction at $54.20 per share" - the price that Mr Musk promised in April.
The apparent win for Twitter sent its shares soaring more than 20% to more than $52 apiece. But the value remained lower than the takeover price, in a sign of lingering investor doubts the deal will go through.
Later on Tuesday, Mr Musk wrote in a tweet: "Buying Twitter is an accelerant to creating X, the everything app".
When Mr Musk first revealed plans to buy Twitter in a $44bn deal, he said he wanted to clean up spam accounts on the platform and preserve it as a venue for free speech.
But the billionaire, a prolific Twitter user known for his impulsive style, balked at the purchase just a few weeks later, citing concerns that the number of fake accounts on the platform was higher than Twitter claimed.
Twitter executives denied the accusations, arguing that Mr Musk - the world's richest person with a net worth of more than $220bn - wanted out because he was worried about the price.
The back-and-forth followed a sharp downturn in the value of technology stocks, including Tesla, the electric car company that Mr Musk leads and is the base of much of his fortune.
The fight, which was scheduled to go to trial 17 October, saw the two sides face off in lengthy court filings, private messages and bitter public spats on Twitter, where Mr Musk has more than 100 million followers.
In one such exchange, Mr Musk responded to Twitter boss Parag Agrawal with an emoji for faecal matter.
Preparation for the trial had ensnarled many of the biggest names in tech, as lawyers for the two companies demanded communications about the deal.
Mr Musk, who could have paid a $1bn break-up fee to walk away, was set to be interviewed ahead of the trial this week.
Some industry watchers, who were taken by surprise by the development, questioned whether the latest twist was a concrete offer or a delay tactic.
Dogecoin, a meme-based cryptocurrency, soared on Friday after Tesla CEO Elon Musk announced that the electric manufacturer will accept it as payment for items. Musk tweeted, "Tesla goods can be purchased with Dogecoin."
His tweet in mid-December stating that such usage of Dogecoin will be permitted on a trial basis propelled the cryptocurrency up more than 20%. After Friday's tweet, Dogecoin, a popular cryptocurrency among retail investors, soared 18 percent to above $0.2.
Musk's tweets about cryptocurrencies, including one in which he referred to it as the "people's crypto," boosted the meme coin, causing it to rise about 4,000 percent in 2021. Musk said this in answer to a query on when the company's electric automobiles will be available in the country.
Last year, Tesla announced its intention to start selling imported automobiles, and the company has been pressing the government to lower import duties on electric vehicles (EVs)...
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