And numerous reports of plotting in the party following the by-election defeat at Glasgow East early on Friday have appeared in the press.
Writing on the Labourhome website, Mr Prescott says he hopes "fellow Labour MPs will take a break too - both from the Westminster bubble and divisive talk of a pointless leadership challenge".
Noting that he had worked with the last three leaders and all cabinet ministers since 1997, Mr Prescott added: "We have, undoubtedly, some very talented men and women.
"But with respect, none of them at the present moment has anywhere near the skills and experience, nationally and internationally, to lead this great party and country as we tackle these unprecedented major global problems."
Mr Prescott left his deputy leader post when Gordon Brown succeeded Tony Blair as prime minister in June 2007. He is set to step down as Hull East MP at the next general election.
He added it had only been a year since Mr Brown became prime minister and "party members and the public will never forgive MPs and others who force Labour to go through another leadership election in less than two years".
BBC political correspondent Adam Fleming said Mr Prescott was talking from the "optimistic wing" of the Labour party but the prime minister would "take some comfort" from his message.
He said an obvious challenger had not come forward but should one emerge most Labour MPs now accepted a general election would need to be called within months to give the new leader a mandate from the public.
Mr Brown started his summer holidays in Suffolk on Saturday
Earlier, Justice Secretary Jack Straw told the BBC he had no plans to mount a leadership bid, despite reports an ally is canvassing support.
Schools Secretary Ed Balls acknowledged it was a testing time for the government but said the party was united on a positive programme for the future.
"The thing we have to show is the strength, the discipline, the backbone as a party to do what the public wants, which is not to turn in on ourselves as Jack Straw warned against, but to show on the big issues - whether it's energy prices, the economy, people's concerns about public services - we can deliver," he said.
Cabinet Office minister Ed Miliband - in charge of drafting Labour's next election manifesto - said the focus needed to be on new policies not political in-fighting.
Speaking from this weekend's National Policy Forum, he said: "That's the things the party members have been talking about. That's what they've been getting excited about and that's the message they want to take out to the country."
But shadow foreign secretary William Hague said Labour was "increasingly losing the authority to govern".
"It's now been defeated in elections in every part of the country and I think they have come to the point where the best thing they could do is call a general election and let a proper government take charge," he said.
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