After tweets emerged that Meta, owned by Mr Zuckerberg, is to release a social media platform to rival Twitter, named "Threads", the SpaceX founder responded: "I’m sure Earth can’t wait to be exclusively under Zuck’s thumb with no other options."
A Twitter user replied jokingly warning Mr Musk to "be careful" after Mr Zuckerberg, 39, has recently won jiu-jitsu tournaments.
Mr Musk, currently the world's second wealthiest person and who turns 52 later this month, wrote back: "I’m up for a cage match if he is lol".
Mr Zuckerberg, whose company owns Facebook, Instagram, and Snapchat, took to his Instagram story on Wednesday with a screenshot of the tweets.
The caption read: "Send me location".
Mr Musk then told him: "Vegas Octagon," an area used for the Ultimate Fighting Championship (UFC) in Las Vegas.
Mr Musk also tweeted: "I have this great move that I call 'The Walrus', where I just lie on top of my opponent & do nothing."
Though many have questioned the seriousness of the dialogue between the pair, a Meta spokesperson told the BBC: "The story speaks for itself."
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