The two billionaires have engaged in a back-and-forth game of both professional and personal one-upsmanship for much of the past 10 years. That rivalry has only increased in recent months, as Mr. Zuckerberg quickly spun up Threads after he witnessed the difficulties Mr. Musk had in his takeover of Twitter.

Threads saw an early burst of enthusiasm, breaking records for the fastest number of downloads for any app in history in its first week. The company said it must now focus on adding new features that keep people coming back as the initial excitement has cooled. That has put pressure on Mr. Musk to continue introducing new features to X, which he has indicated he wants to position as an “everything app” akin to those popular in China, Japan and South Korea.

Their brinkmanship came to a head in June, when Mr. Musk, in a reply to someone on Twitter, said that he would fight Mr. Zuckerberg in a physical cage match. The next day, Mr. Zuckerberg confirmed he was more than willing, and the two men entered into talks presided over by Dana White, president of the Ultimate Fighting Championship. It seemed as if the fight “might actually happen,” Mr. Musk later said in an online discussion.

But those talks appear to have stalled in recent weeks, even as Mr. Musk occasionally joked online that he was training for the match with friends. In response, Mr. Zuckerberg posted photos of his own training regimen with professional Brazilian jiu-jitsu fighters. In July, Mr. Zuckerberg posted a photo of himself shirtless after a training match, boasting a newly chiseled physique and affirming his zeal for the sport.

A spokesperson for Meta declined to comment. Mr. Musk did not respond to a request for comment.

On Sunday, Mr. Zuckerberg said that Mr. Musk’s proposal to stream the fight on X did not seem prudent, suggesting instead that they do it on a platform that was “more reliable” and able to directly raise money for charity. (Mr. Musk’s platform suffered technical issues and glitches over the past year, including problems when he livestreamed his appearance with Gov. Ron DeSantis of Florida, who used Twitter to announce his candidacy for president.)