Fifa is facing calls to re-run the bidding process for the 2022 World Cup after the Sunday Times published emails by disgraced former executive committee member Mohamed bin Hammam.
The paper claims the emails show payments were made to officials as part of a campaign to win the 2022 tournament for Qatar.
England lost out in the contest for the 2018 World Cup, which was run simultaneously with the 2022 process, when Fifa officials voted in 2010.
Speaking at an event in Newark ahead of this week’s by-election, Mr Cameron said: “There is an inquiry under way, quite rightly, into what happened in terms of the World Cup bid for 2022. I think we should let that inquiry take place rather than prejudge it.
“My memories of that bidding process are, as I’ve said earlier, not happy ones in terms of the way the whole thing was arranged and the role of Fifa and the rest of it.
“Let’s let the inquiry take place.
“As for the future of the World Cup in 2018 (in Russia), generally speaking we should try to keep sport and politics separated. I think we should use that as a rule. In extremis there are occasions - Zimbabwe, obviously South Africa - there are occasions where it is right to not take part in a sporting occasion but on the whole we should try and keep them separate.”
Earlier, a Yorkshire MP and former Sports Minister led calls to re-open bidding. Gerry Sutcliffe, who represents Bradford South and served as Minister for Sport and Tourism under Gordon Brown’s government, to call for Fifa to “clean up its act”.
Mr Sutcliffe said: “My view is that if the claims from The Sunday Times stand up then there should be a re-run of the whole bidding process.
“There have been concerns for a number of years in the world of football about the way Fifa is run and the corruption.
“There is already an inquiry from (Fifa’s chief investigator) Michael Garcia and he should be looking in to these Sunday Times claims as part of that investigation.
“It is time Fifa cleaned up its act.”
Mr Cameron said today: “David Beckham has been in my thoughts today because David Beckham, Prince William and I were the team that tried to win for Britain the 2018 World Cup.
“I’ll never forget the meetings we went to, the lobbying we did. It was like no other election I’d been involved in because every single person we met, whether it was the head of the FA in this part of the world or that part of the world, they all said ‘Yes, of course we’re going to vote for England to host the World Cup’ and then they voted completely the other way.
“We ended up, I think, with one vote and I’ll always remember Beckham saying to me afterwards, he said ‘I can cope with being lied to but I can’t cope with people lying to the Prime Minister and the future king’.
“That actually says everything about what a great man David Beckham is.”
The Prime Minister’s official spokesman said at a briefing in Westminster: “It’s absolutely right that there is a very thorough and detailed investigation of these allegations.”
He said they are “very serious reports” about the Qatar tournament, following an investigation published in the Sunday Times.
Deputy Prime Minister Nick Clegg said: “These are shocking allegations about the bidding process for 2022 World Cup. If proven true, Fifa must rerun the contest fairly and openly.”
The organisers of the Qatar 2022 World Cup are to meet international football’s chief ethics investigator amid calls for the Gulf state to be stripped of its right to hold the tournament if corruption allegations over the bidding process are proven.
US lawyer Michael Garcia is due to meet officials from the Qatar bid in Oman after fresh allegations about payments being made to African federations to win support for the successful 2022 campaign.
The meeting comes as senior figures, including Football Association chairman Greg Dyke and former attorney general Lord Goldsmith, said the bidding process would have to be re-run if the corruption claims are shown to be true.
The Qatar bid committee has vehemently denied any wrongdoing and said it would “take any steps necessary” to defend the process.
The allegations come as the results of an inquiry by Mr Garcia are awaited into the 2018 and 2022 World Cup bidding races. He is expected to give his conclusions to Fifa’s ethics committee later this year.
Mr Dyke told BBC Radio 5 Live: “Fifa has got to sort this out. Fifa has got to do their investigation and either it happened or it didn’t happen.
“If it happened and there was corruption involved, then clearly the process has to be re-run.”
Lord Goldsmith, appointed a member of Fifa’s independent governance committee set up to look at ways of reforming the governing body, told BBC Radio 4’s Today programme the decision to award Qatar the World Cup in 2022 should not stand if it is proved it was due to “bribery and improper influence”.
The Sunday Times reported that it has gained access to millions of emails and documents which have highlighted payments made by disgraced former Fifa executive committee member Mohamed bin Hammam.
The newspaper claims the documents show Bin Hammam made payments to football officials as part of a campaign to win support for Qatar’s 2022 World Cup bid.
In a statement, Qatar’s bid committee said Bin Hammam had no association with it and it was co-operating with Mr Garcia’s investigation.
The statement read: “The Qatar 2022 bid committee always upheld the highest standard of ethics and integrity in its successful bid to host the 2022 Fifa World Cup.
“In regard to the latest allegations from the Sunday Times, we say again that Mohamed bin Hammam played no official or unofficial role in Qatar’s 2022 bid committee. We vehemently deny all allegations of wrongdoing.”
Issa Hayatou, president of the African football confederation (CAF) and a Fifa vice-president, issued a lengthy statement denying a number of allegations made by the Sunday Times.
The statement called the allegations “fanciful” and “ridiculous”, and added: “Mr Hayatou has never received any money from Mr Bin Hammam, the Emir of Qatar or any member of the Qatar 2022 bidding committee.”
He said he was not aware of any payments being made by Bin Hammam to African federations, nor had he accepted any flights or privileges from Qatar 2022.
Fifa president Sepp Blatter is also facing calls to step down over the scandal. Mr Dyke refused to say whether the FA would back him for re-election, only that the world governing body needs to find conclusive answers to the allegations.
Asked about whether the FA would support Mr Blatter, who is expected to announce next week that he is to stand again for election in 2015, Mr Dyke added: “That’s a decision that the FA will take at some stage.
“You have got to separate Sepp Blatter from this. Sepp Blatter, as I understand it, didn’t vote for the World Cup in Qatar.
“I wouldn’t give you that answer at the moment. The FA is going to make that decision.”