Twitter is nearing a deal to sell itself to Elon Musk, two people with knowledge of the situation said, a move that would unite the world’s richest man with the influential social networking service.

Twitter’s board was negotiating with Mr. Musk into the early hours of Monday over his unsolicited bid to buy the company, after he began lining up $46.5 billion in financing for the offer last week, said the people, who spoke on the condition of anonymity because they were not authorized to discuss confidential information. The two sides were discussing details including a timeline to close any potential deal and any fees that would be paid if an agreement were signed and then fell apart, they said.

The discussions followed a Twitter board meeting on Sunday morning to discuss Mr. Musk’s offer, the people said. Obtaining commitments for the financing was a turning point for how the board viewed Mr. Musk’s bid of $54.20 a share, enabling the company’s 11 board members to seriously consider his offer, the people said.