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| Elon Musk and Mark Zuckerberg agree to hold cage fight |
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 jiu-jitsu 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.
Elon needs to sign a waiver because Zuck will completely destroy him. https://t.co/qeazJGmi0M
— ST (∞,∞) 🏴☠️ (@seyitaylor) June 22, 2023
choose your fighter pic.twitter.com/9cATjb43rh
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.
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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.
