Downing Street insiders blame such stories on allies of Gordon Brown, the Chancellor, whose insatiable desire to succeed his friend occasionally gets the better of him. Mr Brown's aides are convinced that he and Mr Blair struck a two-point agreement in 1994 when John Smith died.
Part one was that Mr Blair would be the modernisers' candidate to succeed him as Labour leader. Part two, according to the Brownites, was that Mr Blair would stand down to give his long-time political partner a chance at the top job.
"We know where all the speculation about Tony standing down during the next parliament comes from," one Number 10 aide said yesterday. "Gordon's pals have been saying it ever since Tony became Prime Minister."
Mr Blair's aides insist there was no such deal, and dismiss the idea that he will quit after five or six years in Downing Street. "The chances of that happening are less than zero," said one.
A Labour source added: "Tony is always focused on the next election. He wants to win and win and win again."
Such comments echo Margaret Thatcher's words as Prime Minister that she would go "on and on and on". All occupants of Number 10 dream of standing down "at a time of my own choosing, when people least expect it", as John Major put it. But most are unceremoniously evicted by the electorate.
Baroness Thatcher intended at one point to do just seven years and regrets not quitting while she was ahead after 10. The cry from the party was "10 more years" but, 18 months later, it committed the regicide that haunts it today.
Supporters of the "Blair to stand down" theory point out that, after six years in power, he would be young enough at 50 to do other things - seeing his children before they leave home and allowing Cherie Booth, his wife, to become a judge.
Mr Blair has undoubtedly aged in two years at Downing Street; his rapidly receding hairline says it all. After the gruelling Kosovo conflict and the stalled Irish peace process, the fresh-faced Bambi who succeeded Mr Major looks drawn and tired. Yet he remains unfazed by power. Friends from outside the political world insist he has not changed.
Recent events have eaten into his normal "ring-fenced family time", but he will catch up soon at Chequers, which has become the sacred family haven at weekends, and then on holiday in Italy.
Perhaps his lack of Labour roots help him to keep his feet on the ground. As John Prescott once told him: "You can walk away from this party, but I never will."
Mr Blair is not one for anniversaries: he was not even aware that today marked his five years as Labour leader until aides pointed it out. He told officials there should be no fanfares to mark the occasion and to issue a "business as usual" message.
For him, that means Prime Minister's questions in the Commons and a high- risk visit to Eddisbury, Cheshire, where there is a by-election in a Tory- held seat tomorrow.
One reason why he has no intention of standing down after the next general election is that, two years into the job, Mr Blair now realises that modernising Britain is much more complicated than changing the Labour Party. His off- the-cuff remark about having "scars on my back" two weeks ago, widely seen as an attack on public sector workers resisting change, was more a cry of frustration at the glacial pace of progress since 1997.
Mr Blair knows it is no longer credible to blame the problems in Britain's schools, hospitals or decrepit transport system on his inheritance from the Conservatives. By now the voters are looking for evidence that Labour has made a difference.
There are growing fears in Downing Street that the spin doctors may fall victim to their own hype over the "extra pounds 40bn" for health and education (most of which was already in the pipeline). No wonder an impatient Mr Blair summons his Cabinet ministers to talk about "delivery".
Such interventions sum up his hands-on approach to government. "President Blair", as the leading Whitehall watcher Peter Hennessy calls him, has sidelined his Cabinet, preferring to take the real decisions on his settee at Downing Street, with the relevant minister - and sometimes Mr Brown - sitting opposite.
The two pivotal relationships in the Government are Blair-Brown and Blair-Prescott. Despite the occasional outbreak of tension, Mr Blair and Mr Brown remain closer than most previous Prime Ministers and Chancellors.
Mr Brown is much more than a Chancellor, being the overlord of domestic policy and the source of much of Mr Blair's strategic vision. Mr Prescott, meanwhile, refreshes the parts of the Labour Party Mr Blair cannot reach; Mr Blair returns the compliment in Middle England.
Close allies insist that, five years on, Mr Blair is much more confident than he was when he made his acceptance speech and held Mr Prescott's hands aloft. They dismiss criticism that he is too cautious on big issues such as the single currency.
"Tony knows exactly where he is going. He is bold on strategy but cautious on implementing it. He listens to advice before deciding how to get there," said one minister.
Political foes admit even a worn-out Mr Blair is a huge asset. "He is a class act," a senior Tory admitted enviously, after watching a rather sweaty Mr Blair field questions on education at a school on BBC2's Newsnight on Monday. "He was knackered but he was still brilliant."
Another Tory said ruefully: "He has the X-factor. You've either got it or not."
The past two months have not been easy for Mr Blair. He had a narrow escape in Kosovo ("We could barely believe it all ended so quickly," one aide admitted). But then came his first electoral defeat at the Euro elections, and his fabled luck also ran out in Ireland.
But the search for peace in Ireland is a reminder that there is plenty for Mr Blair to do; the battle for lasting change has barely begun. As the party's new membership cards, marking its centenary, say rather cheekily: "The next 100 years start here."