The Tesla boss recently announced his intention to take full control of the social media giant, and it would seem that his planned purchase is now in its final stages.
Earlier this year, Musk became the largest shareholder in Twitter after taking a 9.2 percent stake in the company.
However, just this month, the 50-year-old went a step further and bid $54.20 per share to take full control.
In the statement announcing the agreement, Musk said: "Free speech is the bedrock of a functioning democracy, and Twitter is the digital town square where matters vital to the future of humanity are debated.
"I also want to make Twitter better than ever by enhancing the product with new features, making the algorithms open source to increase trust, defeating the spam bots, and authenticating all humans. Twitter has tremendous potential – I look forward to working with the company and the community of users to unlock it."
Bret Taylor, Twitter's Independent Board Chair, said: "The Twitter Board conducted a thoughtful and comprehensive process to assess Elon's proposal with a deliberate focus on value, certainty, and financing.
"The proposed transaction will deliver a substantial cash premium, and we believe it is the best path forward for Twitter's stockholders."
Twitter CEO Parag Agrawal said: "Twitter has a purpose and relevance that impacts the entire world. Deeply proud of our teams and inspired by the work that has never been more important."
When Musk shared his ambition of buying Twitter, many thought it to be another grandiose claims made by Musk that would come to nothing.
Despite him being the richest man on the planet, much of his wealth is tied up in stocks, so plenty of experts were quick to question how he was actually going to pay for the site.
However, Musk last week released a financing plan, which included the backing of Morgan Stanley and other lenders, offering $13bn (£10.2bn) in debt financing as well as a further $12.5bn (£9.8bn) in loans against his stock in Tesla.
Musk himself also put up $21bn (£16.49bn) through equity financing.
Since he declared his interest, the SpaceX boss has already proposed a couple of major changes.
Earlier this month, he suggested transforming Twitter's headquarters in San Francisco into a homeless shelter.
He also recently told TED founder Chris Anderson what he hopes to achieve with his takeover.
Anderson asked: "So Elon, a few hours ago you made an offer to buy Twitter, why?"
To which Musk replied: "I think it's very important that there will be an inclusive arena for free speech."
"Twitter has become kind of the de facto town square. It's really important that people have the reality and the perception that they're able to speak freely, within the bounds of the law."
He then went on, explaining how he would like to see greater transparency within the site.
"One of the things I think Twitter should do is open source the algorithm and make any changes to people's tweets, if they're emphasised or de-emphasised, that should be made apparent, so you can see that that action has been taken.
So, you know, there's no behind-the-scenes manipulation either algorithmically or manually."
Featured Image Credit: Alamy