World football governing body Fifa is under growing pressure after weekend claims that payments were made to officials as part of a campaign to win the 2022 tournament for Qatar.
Mr Clegg, the Sheffield Hallam MP, said: “These are shocking allegations about the bidding process for 2022 World Cup. If proven true, Fifa must rerun the contest fairly and openly.”
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, Prime Minister David 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.”
Mr Cameron earlier told staff at Knowhow in Newark: “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.”
Calls to rerun the 2022 process were endorsed by football figures including Football Association chairman Greg Dyke.
He said: “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.”
Former attorney general Lord Goldsmith, appointed a member of Fifa’s independent governance committee set up to look at ways of reforming the governing body, said 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.