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".
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."