Mr Clarke's comments had come amid mounting criticism of the prime minister's performance on issues such as Gurkha settlement rights and MPs' expenses, with opposition parties claiming his authority had been fatally undermined.
Responding to Mr Clarke's comments, Mr Prescott said: "Charles, if you are ashamed to stay in the party it's obvious what you should do, isn't it?"
'Get campaigning'
Dismissing the critics of Mr Brown he said: "They are the same people who crawled out of the woodwork last September, told us they were finished, Gordon should go.
"Clarke actually did apologise later and said he thought he'd got it wrong.
"Now he's back at it again. I just say to them, for God's sake stop complaining and get campaigning," he said.
BBC political correspondent Ross Hawkins said it was a clear message from Mr Prescott, who was unimpressed with Charles Clarke's intervention.
Another former home secretary, David Blunkett, has also warned the government appeared to have lost its "political antennae" and needed to avoid further "self-inflicted wounds".
Mr Prescott said that he had been urging Mr Blunkett to join him on his "Prescott Express" campaigning bus.
"I asked him to come on the campaign and he said he couldn't because he was going to a football match," he said.
Justice Secretary Jack Straw said the party had to avoid getting dragged into "narrow arguments" and "rekindle the zeal for power" if it was to win a fourth term in office.
"Every minute people spend talking down our chances of winning the next election is a moment they could be spending working to win the next election," he said in a message to party members in Blackburn to mark his 30th anniversary as an MP.
Meanwhile, Lord Ashdown has said senior Labour figures have discussed defecting to the Liberal Democrats if the party loses the next election.
In an interview with the Daily Telegraph, the former Lib Dem leader indicated some in Labour were concerned about a lurch to the left in the event of a poll defeat.
But Labour sources said talk of defections, rather than discontent, was overblown.
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