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.
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Broadcaster and fight sports journalist Nick Peet told the BBC's World at One programme that Dana White, the president of the UFC, would be "licking his lips at the possibility of putting this fight on".
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 us stone in weight on him.
"But unfortunately Mr Musk has got no training whatsoever. Even though Zuckerberg's only been training Brazilian jiu-jitsu 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!"
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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, Philadephia and Baltimore. This has yet to materialise.
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 favour of his resignation in a poll he ran.
He had already said he wanted to step back as chief executive in November 2022, however.
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Tech rivalry
Earlier this month, Meta showed staff plans for a text-based social network designed to compete with Twitter, sources told the BBC.
It could allow users to follow accounts they already follow on Instagram, Meta's image-sharing app.
It could potentially allow the company to bring over followers from decentralised platforms such as Mastodon.
A Meta spokesperson confirmed to the BBC that the platform was in development.
The text-based network - which has a working title of P92 - could turn out to be a greater rival to Elon Musk's Twitter than either BlueSky or Mastodon.
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