Elon Musk "isn't serious" about holding a cage fight and "it's time to move on", Mark Zuckerberg has said.
In a post on his social media site Threads, the Meta boss said he had offered Mr Musk "a real date" but the rival entrepreneur had made excuses.
Mr Musk had earlier on Sunday suggested on his own messaging site X, formerly known as Twitter, that he was willing to fight as early as Monday.
The billionaires agreed to the bout in June, sparking huge media attention.
But despite egging each other on for months, the rivals have yet to secure a date, raising doubts the fight will ever go ahead.
The war of words was complicated by the launch of Threads in July, with the rival messaging app to X attracting more than 100 million sign-ups in less than a week.
That number has fallen back, and X remains comfortably ahead with around 350 million users – but Mr Musk has threatened to sue Facebook for "unlawful misappropriation of Twitter's trade secrets".
Meta denies the claims.
Last week, Mr Zuckerberg said he had proposed 26 August for the cage fight. Then in a twist on Friday, Italy's culture minister said that he had spoken to Mr Musk about hosting the showdown in the country as a charity event.
Mr Musk suggested it would have "an ancient Rome theme".
On Sunday, however, the Meta boss posted on Threads: "Elon won't confirm a date, then says he needs surgery, and now asks to do a practice round in my backyard instead.
"If Elon ever gets serious about a real date and official event, he knows how to reach me. Otherwise, time to move on. I'm going to focus on competing with people who take the sport seriously."
Responding on X, however, Mr Musk called the Meta boss a "chicken".
Earlier, Mr Musk had posted a text message exchange on the messaging platform purportedly between himself and Mr Zuckerberg.
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In it he tells the Meta boss that he will be in Palo Alto, home to Meta's headquarters, on Monday and that the fight could be held in Mr Zuckerberg's Octagon, which is the eight-sided ring in which cage fights are held.
Mr Musk continued: "I have not been practicing much, apart from a brief bout with Lex Fridman [the computer scientist and podcast host] today.
"While I think it is very unlikely, l given our size difference, perhaps you are a modern day Bruce Lee and will somehow win."
Elon Musk, 52, and Mark Zuckerberg, 39 are two of the world's most high-profile technology billionaires.
The bizarre idea to fight each other started in June, when Mr Musk tweeted that he was "up for a cage fight" with Mr Zuckerberg.
The Meta boss, who already has mixed martial arts (MMA) training and has recently won jiu-jitsu tournaments, simply responded with "send me location".
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