Mr Blair is said to have issued the threat last week after Mr Brown recalled the way Margaret Thatcher was forced out by her own party in 1990. "Remember that when Margaret Thatcher left, it was unstable, it was disorderly and it was undignified," the Chancellor told GMTV.
According to allies, the Prime Minister viewed Mr Brown's remarks as a "naked threat" to force him out of office. One said: "It was communicated to Gordon that if Tony was deposed, he and his supporters would not support Gordon as the next leader."
Blairites deny they plan to put up a rival candidate against Mr Brown, the overwhelming front-runner to succeed Mr Blair. Yesterday friends of John Reid, the Blairite Home Secretary, dismissed reports that he was canvassing support for a leadership bid.
"Tony is not going to run a candidate against Gordon," one Downing Street insider said. "But if he is deposed, he will not support him." If Mr Blair carried out such a threat, aides said, it would make it harder for Mr Brown to win votes in Middle Britain.
The private warning to Mr Brown is at odds with Mr Blair's public comments. A week ago, asked whether the Chancellor was his chosen successor, he replied: "Of course he is. When have I ever said ... different?"
There was growing speculation yesterday that Mr Blair would stand down in the summer of next year, allowing Mr Brown to be crowned Labour leader and Prime Minister at the party's annual conference in September. However, allies of Mr Blair suggested that he would leave Downing Street more quickly if Mr Brown threw his weight behind the reforms he wants to push through before departing and signed up to a strongly New Labour agenda for the future. Mr Blair wants to do everything in his power to keep Labour committed to reforms after he quits, believing that is the best way to ensure a fourth successive election victory.
Supporters of Mr Brown say he is happy to agree a five-year programme of work with Mr Blair, believing this would help a smooth transition and to reassure voters he is committed to New Labour. But he deeply resents suggestions from ultra-Blairites that he is anti-reform, saying this plays into the hands of the Tories.
A source close to Mr Brown denied as "completely untrue" the claim that Mr Blair had threatened to withdraw his support for him as his successor. He said: "It's just the same old divisive briefing from the same old divisive outriders, who now seem completely out of the loop. Haven't they learnt that what the Labour Party wants and the country needs is a unified process where Tony and Gordon work together to achieve a stable and orderly transition, not these factional attempts to disrupt that process? Tony has told these people to shut up and work for the good of the party, and they should listen to him."
Brownites said the Chancellor had hoped senior Labour figures had "moved on" after he struck an agreement with Mr Blair on pensions policy last week, which showed how they could work together.
Mr Blair is trying to keep the media spotlight on policy and his reforms by setting out the challenges facing every Whitehall department in a letter to the relevant Cabinet minister.
Yesterday he told the Health Secretary, Patricia Hewitt: "We must continue the pace of reform by improving the choices that patients have within the NHS. We know this empowers patients and, through links with payment by results, drives improvement and value for money. We must therefore continue to expand the effect of choice and payment by results."
The Prime Minister said the NHS trusts which had run up persistent deficits "must use this year to make in-roads into their long-term problems."
Today, Mr Blair will show his commitment to further reform when he launches a consultation exercise, entitled "Let's Talk," about Labour's future policies, involving the public and interested groups, as well as the party. The first meeting tackles public services.
Mr Blair will come close to admitting that, nine years after winning power, the criminal justice system is not up to the job. He will say: "I believe people want a society without prejudice but with rules; rules that are fair; that we all play by; and rules that when broken carry a penalty. The truth is most people don't think we have such a society.
"The problem of crime can be subject to lurid reporting or undue focus on terrible, but exceptional, cases. But even allowing for this, the fundamental point is valid. Despite our attempts to toughen the law and reform the criminal justice system ... [it] is still the public service most distant from what reasonable people want."
The Prime Minister will argue that modern social democracy must find answers to new questions or fall back. "We need this debate to be open, frank and engage public as well as party. The most effective politics today is not tribal. It is issues-based. And we should play our part with confidence," he said.
Mr Blair will launch the initiative in London with John Prescott, the Deputy Prime Minister, and Hazel Blears, the Labour Party chairman. Ms Blears will say: "As we update and renew New Labour, this process is crucial. It will allow us to step back from the fray, think about our policies and consider how best to match our progressive values to the demands and needs of the British people."
An ICM poll for the BBC's Politics Show found that most voters in England believed Scottish MPs should be barred from becoming Prime Minister. Across the UK, 52 per cent of people said it was wrong for an MP north of the border to become Prime Minister now Scotland has its own Parliament, while 45 per cent believed it was right. Only 20 per cent of those polled in Scotland opposed a Scottish MP becoming Prime Minister.
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