Twitter Inc. is in the final stretch of negotiations about a $43 billion sale to Elon Musk that could rank as one of the biggest-ever leveraged buyouts of a listed company, people with knowledge of the matter said.

The social media company is working to hammer out terms of a transaction and could reach an agreement as soon as Monday if negotiations go smoothly, according to the people, who asked not to be identified because the information is private. Musk is lining up partners for the acquisition and continues to speak to potential co-investors, one of the people said.

Discussions between the billionaire Tesla Inc. founder and Twitter’s board about a takeover at $54.20 per share continued overnight into the early hours of Monday, the people said. Shares of Twitter jumped as much as 6.2% in pre-market U.S. trading Monday, hitting as high as $51.98.

Twitter started warming up to a potential deal after the billionaire Tesla Inc. founder revealed a financing plan for the unsolicited bid that included backing from Morgan Stanley and other institutions. The situation is fluid, and talks could drag on longer or fall apart, the people said.

Representatives for Twitter and Musk didn’t immediately respond to requests for comment. The Wall Street Journal reported earlier that the parties could reach an agreement as soon as this week, citing unidentified people.

—With assistance from Maxwell Adler and Scott Deveau.