Former Labour deputy leader Lord Hattersley has called on Tony Blair to quit as prime minister in September.
Lord Hattersley says Gordon Brown should be the next Labour leader
He said any tenure in Downing Street beyond this year's Labour conference would damage the party.
Last week left-wing backbencher John McDonnell, Hayes and Harlington MP, said he would run for the Labour leadership when Tony Blair stands down.
But Lord Hattersley said Chancellor Gordon Brown was the "ideal" candidate to take over the reins of the party.
"The prime minister ought not to announce he's going at party conference - he ought to go at party conference," the Labour peer told GMTV's Sunday Programme.
Mr Blair said he did not want constant speculation over the timing of his resignation.
"I'm not sitting there, you know, obsessing the entire time about when the precise date is and all the rest of it. Get on with the job, that's what the public want and we are," he told BBC One's Politics Show.
He said the Labour Party had a "very strong sense of direction".
Mr Hattersley said Mr Blair should point to Labour's achievements since coming to power in 1997, pass the leadership baton to his successor and "thank the party for what they've done".
"If we did that I think he'd go out on a high note in the party, I think that'll ripple out through the country.
"The longer he stays on the more damaging it is for him as well as the party in my view."
He also said that if Mr Blair stayed until 2007, it would be "impossible to put new life" into the party.
The deputy leader of the Labour party from 1983 to 1992, Lord Hattersley said the chancellor was the best successor to Mr Blair.
Lord Hattersley wants Tony Blair to stand down as PM by September
"I think Gordon Brown increasingly becomes the ideal next leader of the Labour party," he said.
"I think what the Labour Party needs to win the next election is a solemn, serious, if you like, son of the manse.
"Somebody with visibly and almost ostentatiously contemptuous of spin who wants simply to say it as it is."
Lord Hattersley also felt the cash-for-honours allegations were "deeply damaging" to Labour.
But the former MP for Birmingham Sparkbrook said be believed Mr Blair and his fundraiser Lord Levy - who was arrested last week but has not been charged - were innocent.
He said he would be "astonished" if either the prime minister or Lord Levy were charged.
He described Mr Blair as "instinctively honest".
Liberal Democrat party president Simon Hughes said: "Roy Hattersley's call for the prime minister to go this year just piles on the pressure.
"Every month that passes looks more and more like the end of the Blair era. The chosen date for going is, of course, a personal decision and principally a Labour Party concern.
"But good government for Britain and our reputation abroad requires a prime minister with a future not a past and a prime minister who commands authority in his party as well as the support of the British people."