The proposed deal, first reported by Bloomberg News, matches the original offer price Mr Musk made to the San Francisco-based microblogging platform in April.
“We received the letter from the Musk parties which they have filed with the SEC. The intention of the company is to close the transaction,” Twitter said on Tuesday.
Twitter shares jumped 22.2 per cent to close at $52.00, while Tesla shares gained 2.9 per cent to $249.44 having dived from $256.16 a share to $245.27 within minutes of the report.
Mr Musk, the world's richest person, agreed this year to acquire Twitter in a $44 billion deal, pledging to pay $54.20 a share. On completion of the proposed deal, Twitter was to be converted into a private company.
The billionaire said at the time that he wanted to introduce new features and make algorithms open source to increase trust, “defeat the spam bots, and authenticating all humans”.
Mr Musk, later on Tuesday, tweeted that "Buying Twitter is an accelerant to creating X, the everything app".
It is not clear what an "everything app" called X would entail, however, Mr Musk did add in a further tweet that "Twitter probably accelerates X by 3 to 5 years, but I could be wrong".
Meanwhile, the Twitter deal faced various headwinds.
On July 8, Mr Musk filed paperwork to terminate the transaction, claiming Twitter did not address inquiries on its “spam” or bot accounts, or provide him with relevant business information.
Twitter chief executive Parag Agrawal hit back at Mr Musk's concerns and said spam accounts accounted for less 5 per cent of users.
The digital platform locks millions of accounts each week that it suspects could be fake, he said.
It was claimed that Mr Musk was looking for a reason to back out of the deal.
On July 12, Twitter filed a legal challenge against Mr Musk for breaching the agreement.
On August 29, hs lawyers sent a letter to Twitter to formally terminate the merger agreement.
In that letter filed with the SEC, Mr Musk's lawyers said allegations made by the microblogging platform's former head of security Peiter Zatko breached the merger agreement and served as a “basis” to end the deal.
Mr Zatko claimed that Twitter was in material non-compliance with legal data privacy obligations, fair trade practice and consumer protection laws and regulations.
Both sides have been preparing for a courtroom showdown, which is set to start on October 17 in Wilmington, Delaware.
Lawyers for Twitter and Mr Musk have issued subpoenas to banks, investors and lawyers involved in the deal.