The billionaire proposed a deal with the social media company that could end the legal fight between him and the company.
Two people who were not authorized to speak publicly said that the arrangement would allow Mr. Musk to acquire the company at a price he agreed to in April.
The potential deal comes after months of disputes that have caused problems for the company, including cratering its share price, demoralizing its employees and scaring advertisers.
A deal at the original price would be a big win for the company. In July, Mr. Musk said that he no longer intended to complete the acquisition because he thought the service was overrun by junk mail.
It was not immediately clear if the offer would be accepted by the company. The social media company's shares spiked after the news broke.
An agreement would mean an end to the uncertainty that is clouding the future of the company. The original deal with Mr Musk was approved by shareholders in September.
In July, the company sued Mr. Musk to force the completion of the acquisition, and was ready to go to court this month. The company argued that Mr. Musk had simply hoped for a lower price after stock market declines had decreased his wealth.
Mr. Musk said that the company was not as valuable as he had thought. A whistle-blower claimed that the company had misled regulators about its security practices as a reason to leave the deal.
A person familiar with the conversation said that Mr. Musk sent a proposal to the micro-messaging service on Monday evening. The proposal was discussed in court. Earlier, the offer was reported.
It's possible that a deal will allow both sides to avoid a messy public trial. Mr. Musk will be deposed in Austin, Texas, on Thursday and Friday.
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