The SpaceX CEO could reportedly take control of the company should a deal be reached smoothly today.
Mr Musk, the richest man on the planet, has reportedly floated a figure of $43billion (£34billion) as his “best and final” offer, according to people familiar with the deal.
The social media company is expected to announce the deal later on Monday after the board meets to recommend the transaction to shareholders.
But the sources did warn that is always possible for a deal to collapse.
Mr Musk already owns over a 9 percent stake in Twitter, but is reportedly set to take over the firm with the help of financial backing from Morgan Stanley and other firms.
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This stake in the company was announced last week on April 9, and made the Tesla boss Twitter’s biggest shareholder.
Mr Musk has in the past expressed that Twitter needs to taken private so it can be used as a platform for free speech and further grow as a company.
This is a breaking story. More to follow.