In a post on the new social media app Threads, Mr Zuckerberg said he had proposed 26 August for the showdown.
Following a post on X in which Mr Musk claimed he was training “throughout the day”, Mr Zuckerberg wrote: “I’m ready today… but he hasn’t confirmed.”
The two became direct competitors in July with the launch of Threads.
Asked by a Threads user whether the fight had been mutually agreed upon, Mr Zuckerberg responded that is was more like “funding secured,” in an apparent reference to posts made by Mr Musk in 2018 when he said the same about plans to take electric car company Tesla into private hands.
That deal never happened and led to Mr Musk paying a $20m (£15.7m) fine to the US financial markets watchdog, stepping back from being Tesla’s chairman and limits put on what he can tweet about Tesla.
Mr Musk cast further doubt about the potential bout, saying that the “Exact date is still in flux”, as he may need surgery on his neck and upper back.
The social media moguls have been egging each other on in recent months, with Mr Musk claiming on Sunday that their fight would be broadcast live on his social media platform X, formerly known as Twitter.
He said that any proceeds from a match would go to veterans charities.
When asked what the point of the bout was by one X user, Mr Musk responded: “It’s a civilized form of war. Men love war.”
Mr Zuckerberg also shot back at the suggestion that the fight would be streamed on X, posting: “Shouldn’t we use a more reliable platform that can actually raise money for charity?”
The stakes are seemingly high after Meta, which also owns Instagram and Facebook, launched Threads in early July, drawing in more than 100 million sign-ups within days.
However, Mr Zuckerberg later said the platform had lost more than half of its users by the end of last month.
Rival social media platform X has faced criticism on several occasions since Mr Musk took over the firm and made a number of changes, such as forcing users to log in to view posts. He also carried out mass firings at the company.