Mr Musk posted a message on his social media platform Twitter that he was "up for a cage fight" with Mr Zuckerberg.
Mr Zuckerberg, the boss of Facebook and Instagram parent company Meta, then posted a screenshot of Mr Musk's tweet with the caption "send me location".
"The story speaks for itself," a Meta spokesperson told the BBC. Mr Musk then replied to Mr Zuckerberg's response with: "Vegas Octagon."
The Octagon is the competition mat and fenced-in area used for Ultimate Fighting Championship (UFC) bouts. The UFC is based in Las Vegas, Nevada.
Mr Musk, who turns 52 later this month, also tweeted: "I have this great move that I call 'The Walrus', where I just lie on top of my opponent & do nothing."
He later tweeted short videos of walruses, perhaps suggesting his challenge to Mr Zuckerberg may not entirely be serious.
He also tweeted: "I almost never work out, except for picking up my kids & throwing them in the air."
Meanwhile, 39-year-old Mr Zuckerberg has already been training in mixed martial arts (MMA) and has recently won jiujitsu tournaments.
Twitter did not provide a statement when contacted by the BBC for comment.
The exchanges have gone viral, with social media users debating who would win the bout, while others have posted memes including mocked up posters advertising the fight.
For example, business consultant Seyi Taylor tweeted: "Choose your fighter" with pictures of the two tech bosses.
Broadcaster and fight sports journalist Nick Peer told the BBC's World at One program that Dana White, the president of the UFC, would be "licking his lips at the possibility of putting this fight on".
He said he thought there was a decent chance it could actually go ahead, "mostly because of Elon Musk and his personality and his eccentric character. His career kind of suggests he's not somebody who willingly steps down."
However, asked who he thought would come out on top, he replied:Zuckerberg all day! He's 12 years younger. He is a lot smaller. I think he's 5ft 7, Elon's probably around 6ft. And Elon's probably got a couple of stone on him.
"But unfortunately Mr Musk has got no training whatsoever. Even though Zuckerberg's only been training Brazilian jiujitsu for 18 months, it wouldn't be difficult for him to take his back, wrap his arms around his neck and give him a good old cuddle and choke him out!"
Mr Musk has a history of making statements that are not serious or which fail to happen.
For example, he told the BBC in April he had made his dog chief executive of Twitter.
In 2017, he tweeted he had "verbal government approval" for a so-called Hyperloop - a kind of train system - to connect New York City with Washington DC, Philadelphia and Baltimore. This has yet to materialize.
In 2018 Mr Musk was forced to step down as Tesla chair by regulators after Tweeting that he intended to take the firm private.
Mr Musk has also made good on some of his pronouncements, including stepping down as Twitter chief executive this year after Twitter users voted in favor of his resignation in a poll he ran.
He had already said he wanted to step back as chief executive in November 2022, however.
You must be logged in to post a comment.