Chancellor Gordon Brown is Tony Blair's "natural successor", former Cabinet colleague Peter Mandelson says.
Will Brown inherit the crown?
In an interview with ex-Downing Street communications chief Alistair Campbell, Mr Mandelson says he "respects" Mr Brown's achievements.
The interview, an apparent bid to end years of suspicion between the men, will be shown on Five on Wednesday.
The tension is thought to date from Mr Mandelson's decision to back Mr Blair over Mr Brown as Labour leader in 1994.
In the interview, former Northern Ireland secretary Mr Mandelson says: "I think that when Tony Blair chooses to stand down as prime minister, or is not elected by the public, I think that Gordon Brown will be his New Labour successor.
"Gordon is a big person, is a big politician with very big ideas, as we've seen in what he's done already in government as chancellor, and he has many more big ideas in him.
"I know that, I recognise it, I respect it."
But any suggestions that Mr Blair might be willing to step down in favour of the chancellor in the near future were dismissed by Commons Leader Peter Hain.
Speculation about the leadership was "just nonsense" and the relationship between the prime minister and Mr Brown was unchanged, he told GMTV's Sunday programme.
He said: "Gordon Brown is a towering figure in the government and has been a hugely impressive chancellor. Obviously Tony is the dominant figure in the government and Gordon is right behind him.
"We have one of the most successful prime ministers that the Labour government has ever had.
"I see, in Tony Blair, somebody who is absolutely determined... who has really got his eye on the ball of winning the next election and driving this government forward."
Meanwhile, Mr Brown's confidante Ed Balls, who has quit his Treasury post to stand for election as an MP, has disclosed how close Mr Brown came to quitting at the end of 1999.
He told the Sunday Telegraph Mr Brown had been tempted to take a job running the International Monetary Fund.
Mr Balls said: "A number of finance ministers contacted him to say that they really wanted him to take seriously the idea of moving over."
But Mr Brown decided to stay in his post because he felt his work at the Treasury was not complete, Mr Balls said.