However, Downing Street has rebutted the claim, saying that it is merely "wishful thinking" on the part of the BBC, which is embroiled in a tit-for-tat fight with Campbell over one of its reporters, Andrew Gilligan, and the so-called "dodgy dossier" on weapons of mass destruction in Iraq.
Andrew Marr, political editor at the BBC, has said that Campbell met with Blair on Thursday and that the pair agreed that if he left before the inquiry into Kelly's death, it could imply that Downing Street had done some wrong.
Marr told Radio 4's 'Today' programme: "I am in no doubt at all that, as we are now, Alastair Campbell is determined to go, that the prime minister knows that and has agreed to that.
"Of course, everybody is human, people can change their minds, but at the moment he is certainly going this year."
The Prime Minister's office has retaliated by saying that the BBC is "fixing on gossip rather than substance".
"The main issue remains that the BBC broadcast unsubstantiated allegations and we stand by our position on this matter, which will be examined by the Hutton inquiry."
Campbell has become a hate figure for much of the British media, portrayed sometimes has having great power over the decisions of the man he works for.
Political editor of the Daily Mirror before going to work in media relations, Campbell has given few interviews about his role as the Prime Minister's top PR man, but this has failed to keep him out of the headlines on a regular basis.
Famously a supporter of first division Burnley football club, he told BBC Sport's website last year: "As rare as it is for me to praise the BBC, you can usually find out what's going on in the world though, thanks to a pretty good website and the World Service."
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