Fifa is facing calls to rerun the bidding competition for the 2022 World Cup after allegations that a former top Qatari football official paid $5m (£2.98m) to win support for the nation's campaign to host the event.

Labour said Qatar should lose the 2022 World Cup and urged those involved to resign if the Fifa corruption allegations detailed in the Sunday Times were true. Clive Efford, the shadow sport minister, said the new revelations "called the governance of football into question".

"No one will have any confidence in a Fifa investigation run by Sepp Blatter," he said. "If these allegations are true then those involved should resign."

Lord Macdonald, the former director of public prosecutions, described Fifa as a "bit of a cesspit" and suggested there was evidence of a "very serious crime" following the reports.

He told Sky News's Murnaghan programme: "The idea of another voting session with all this money sloshing around is almost too much to bear. But on the other hand, if I can pretend to be a prosecutor again for a minute, this is evidence of a very serious crime. And the fact that the allegation is that they used dollars, US dollars, means that the justice department in Washington has jurisdiction over this … the United States of course are in the World Cup finals. If the justice department started to take an interest in this, I think Fifa would feel the heat very, very quickly."

The Sunday Times said it had obtained millions of emails and other documents relating to alleged payments made by Mohamed bin Hammam, the then Fifa executive member for Qatar. The paper said Bin Hammam, also the former Asian Football Confederation (AFC) president, used slush funds to pay out the cash to top football officials to win a "groundswell" of support for Qatar's World Cup bid.

John Whittingdale, chairman of the House of Commons culture, media and sport select committee, called for the bidding competition to be held again and said it would be impossible for Fifa to brush aside the revelations.

"It is a further demonstration of the need for a complete change in the way that Fifa operates but also that there is now an overwhelming case that the decision as to where the World Cup should be held in 2022 should be run again," Whittingdale told the paper.

The sports minister, Helen Grant, said: "These appear to be very serious allegations. It is essential that major sporting events are awarded in an open, fair and transparent manner."

The allegations come less than two weeks before the start of the World Cup in Brazil and bring fresh scrutiny on the 2010 vote, which gave football's biggest tournament to the tiny desert state. The process is under investigation by Fifa's independent ethics prosecutor, Michael Garcia.

Qatar has also come under fire for its labour laws, in the wake of an international outcry over conditions for migrant workers before the 2022 World Cup, which followed a Guardian investigation into workplace abuse in the Gulf state.

Bin Hammam is no longer a committee member of world football's governing body after being caught up in a corruption scandal surrounding his failed campaign for its presidency in 2011. The Sunday Times alleged that he exploited his position as an executive committee member to help to secure votes from key members of its 24-man ruling committee that helped Qatar win the right to host the World Cup. Qatar defeated bids from the US, Japan, South Korea and Australia.

According to the newspaper, Bin Hammam used 10 slush funds controlled by his private company and cash handouts to make dozens of payments of up to $200,000 into accounts controlled by the presidents of 30 African football associations who influenced how Africa's four executive members would vote. He also allegedly hosted lavish junkets for these African officials at which he handed out almost $400,000 in cash.

Last month, Blatter said it had been a mistake to choose Qatar for the World Cup, forcing Fifa to try to limit the damage. "Yes, it was a mistake of course, but one makes lots of mistakes in life," said Blatter, Fifa's president, in an interview with the Swiss broadcaster RTS. "The technical report into Qatar said clearly it was too hot but the executive committee – with a large majority – decided all the same to play it in Qatar."

Blatter, who is standing for another term as president in 2015, is believed to have voted for the USA to host the 2022 World Cup, while his prospective rival for the presidency, Uefa's Michel Platini, voted for Qatar and has been closely linked with the plans for the 2022 tournament.

The Fifa inspection team ranked Qatar as the only "high-risk" option overall, yet it was still chosen by 14 of the 22 voting members of the executive committee in December 2010. The Fifa president said it was now "probable" that it would be played in the winter rather than the summer due to the heat. Blatter insisted, however, that Qatar, which spent huge sums on ambassadors and development programmes, had not "bought" the World Cup.

Jim Boyce, Fifa vice-president, said he would be in favour of rerunning the vote if allegations that widespread corruption was involved in the bid were proved. Boyce, who was not on the executive committee of the world governing body at the time of the vote, said Garcia, would have to widen his investigation.

Boyce told Radio 5 Live's Sportsweek programme: "As a member currently of the Fifa executive committee, we feel that any evidence whatsoever that people involved were bribed to do a certain vote, all that evidence should go to Michael Garcia, whom Fifa have given full authority to.

"If Garcia's report comes up and his recommendations are that wrongdoing happened for that vote for the 2022 World Cup, I certainly as a member of the executive committee would have absolutely no problem whatsoever if the recommendation was for a revote."

Anna Soubry, minister for defence personnel, welfare and veterans, said: "Somebody somewhere has got to get a serious grip on Fifa about the way that they run these competitions."

Jim Murphy, the shadow international development secretary, also called for a rethink if the allegations were found to be true.

He told BBC Radio 5 Live's Pienaar's Politics: "If these allegations and the contents of the emails that the Sunday Times now has turn out to be true there can be no question about this. The thing wasn't done fairly, it wasn't done openly and it would have to be cancelled and rerun entirely. The building that is happening in Qatar should be paused and they should have a fair and open competition."

Qatar's World Cup officials said the bid committee had "always upheld the highest standard of ethics and integrity".

The country's Supreme Committee for Delivery and Legacy said: "Mohamed bin Hammam played no official or unofficial role in Qatar's 2022 bid committee. As was the case with every other member of Fifa's executive committee, our bid team had to convince Mr Bin Hammam of the merits of our bid.

"We are cooperating fully with Mr Garcia's ongoing investigation and remain totally confident that any objective inquiry will conclude we won the bid to host the 2022 Fifa World Cup fairly.

"Following today's newspaper articles, we vehemently deny all allegations of wrongdoing. We will take whatever steps are necessary to defend the integrity of Qatar's bid and our lawyers are looking into this matter. The right to host the tournament was won because it was the best bid and because it is time for the Middle East to host its first Fifa World Cup."