The rival billionaire tech bosses seemingly agreed to a brawl in June when Musk tweeted that he was “up for a cage fight”.
Meta boss Zuckerberg, who manages Facebook and Instagram, took a screenshot of Musk’s tweet, replying “send me location”.
Last week, Ultimate Fighting Championship (UFC) boss Dana White said he was in talks with officials in Italy to use the Colosseum in Rome as the location of the fight.
But on Sunday Zuckerberg said on his social media platform Threads that he didn't believe the fight would happen with Musk, who owns X, formerly known as Twitter.
Zuckerberg said: ”I think we can all agree Elon isn’t serious and it’s time to move on.
“I offered a real date. Dana White (UFC boss) offered to make this a legit competition for charity.
“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.”
Musk had appeared to suggest the fight would be held in an “epic location” in Italy.
He had outlined streaming options and claimed he had spoken to Italian prime minister Giorgia Meloni about the bout.
Zuckerberg is trained in mixed martial arts, posting about completing his first jiu jitsu tournament earlier this year.
Last week, White said he was in talks with officials in Italy to use the Colosseum, a Unesco World Heritage site, as the location of the fight.
Talking on Mike Tyson's podcast, White said he believed the fight would generate $1bn in revenue.
The UFC boss said: "It’s real, it's real, they do want to fight. We were literally talking about the Colosseum for that fight.
"I actually did have a meeting with the minister of culture's team in Italy to actually have that fight in the Colosseum, if it would happen.
"It would be MMA [mixed martial arts]. We're talking. No steroids, no steroids allowed; these guys would be drug tested."
Yahoo News UK investigates why the pair agreed to fight and when it may take place:
When is the Elon Musk v Mark Zuckerberg fight?
Earlier this month, Musk posted on X that the platform will live-stream his fight against Zuckerberg.
He wrote: "Zuck v Musk fight will be live-streamed on X. All proceeds will go to charity for veterans."
Musk also revealed he was training for the fight.
"Am lifting weights throughout the day, preparing for the fight," he wrote.
"Don't have time to work out, so I just bring them to work."
But later that day, Zuckerberg hit back, saying he had already offered a date later this month for the fight, but that he didn't think Musk would step into the ring with him.
"I'm ready today," Zuckerberg posted on his company's new social media app, Threads.
"I suggested 26 August when he first challenged, but he hasn't confirmed. Not holding my breath."
But Musk said the "exact date is still in flux", saying he may need surgery on his neck and upper back.
Zuckerberg also jibed that X might not be the best platform to stream the fight, writing: "Shouldn't we use a more reliable platform that can actually raise money for charity?"
Why are Musk and Zuckerberg fighting?
Musk and Zuckerberg are now direct tech rivals after the latter launched Threads last month. Musk concluded his acquisition of Twitter last October.
In June, Musk posted on what was then Twitter that he was “up for a cage fight” with Zuckerberg.
Zuckerberg agreed to the challenge by taking a screenshot of Musk’s tweet and replying “send me location”.
In response, Musk suggested the “Vegas Octagon”, in reference to the venue used for the Ultimate Fighting Championship.
Musk said he had a “great move” to show off to Zuckerberg and noted that his workout regime consists of spending time with his children.
“I have this great move that I call 'The Walrus', where I just lie on top of my opponent and do nothing,” he posted.
He also added: “I almost never work out, except for picking up my kids and throwing them in the air.”
Zuckerberg would appear to be in good shape for the challenge after winning his first jiu-jitsu competition recently.
After being asked what the point of the fight was by one X user, Musk replied: "It's a civilised form of war. Men love war."
More than 100 million people signed up for Threads within days of its launch, but Zuckerberg said later that the platform had lost more than half of its users.
Musk has been dogged by criticism since he took over Twitter and overhauled the site, eventually changing its name to X.
Watch: 'X' sign removed from Twitter's former headquarters