The Deputy Prime Minister wants Mr Blair to announce a firm timetable for his departure at the Labour Party conference in a year's time. He hopes that would enable Gordon Brown to take over in the spring or summer of 2007.
Mr Prescott warned Mr Blair that the Labour Party would not tolerate him staying in No 10 without making clear when he will stand down.
The revelation came as party members gathering in Brighton for this year's conference focused on the uncertainty over the timing of Mr Blair's departure from Downing Street.
As allies of the Chancellor urged the Prime Minister to set out a timetable for his plans to step down, Mr Blair in a television interview yesterday refused to divulge details of his departure.
Mr Prescott's warning is revealed in a biography of the Deputy Prime Minister by Colin Brown, The Independent's Deputy Political Editor. It discloses Mr Prescott's anxiety about Mr Blair's intention to serve a "full term" after he won his third election in May, even though he had promised to quit before the next election.
He warned that if Mr Blair prevaricated until the autumn of 2008 Mr Brown could be left with little more than six months as Prime Minister to prepare for the next general election in the spring of 2009.
The book says: "Prescott forcefully relayed the message to Blair that he did not think the party would allow him to carry on, regardless of the election result, for another three or more years."
Allies of the Chancellor confirmed yesterday that they want Mr Blair to announce a timetable by next year's conference to give Mr Brown time to establish himself as Prime Minister before the next general election.
Today Mr Brown will maintain a public truce with Mr Blair by promising that he would not abandon "New Labour" or vacate the political centre ground if he succeeds him.
But some Blair aides continue to hint that he intends to continueon until 2008 in an attempt to complete his public service reforms and cement his political legacy. Tessa Jowell, the Culture Secretary, told GMTV's Sunday programme that Mr Blair could remain in Downing Street until 2008 or 2009. But that would provoke a new rift with Mr Brown.
In an interview with The Independent, Ed Balls, the Labour MP and one of the Chancellor's closest allies, questioned whether Mr Blair was now the right man for the job of renewing the party so that it could address the long-term challenges facing Britain.
He said: "The Labour Party has got to decide what we need to do in relation to long-term policy decisions. That is the issue, not individual ambition or legacies. That is to belittle the challenges and the changes to the party required to meet those challenges".
Although some Blairite cabinet ministers have anointed Mr Brown as Mr Blair's successor in recent days, the Brown camp suspects that the move is designed in part to ensure that the Chancellor allows the Prime Minister more time in Downing Street. Mr Balls said: "The issue is going to be decided by the Labour Party and is not going to be decided by MPs, however junior, and cabinet ministers, however senior. But the party must focus also on how we meet the challenges for the longer term."
In a BBC television interview yesterday, Mr Blair refused to reveal his plans, saying he had not yet make up his mind when he wanted to leave No 10. But he hinted that he will announce a timetable in due course.
The Prime Minister said: "I've spoken about this as much as I want to speak at the moment. There will come a point in time when I'm delighted to speak about it but for the moment I think, you know, we've just won a third election victory, we should concentrate on setting out the agenda for the country."
The biography of Mr Prescott says that tensions between Mr Blair and Mr Brown could reopen if the Chancellor's desire to take over in good time before the next election is frustrated. It quotes a Brown supporter as saying: "[The Blair camp] want to be reasonable, and that is the case at the moment. If that changes, then our attitude will change."
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