The European Union is to make reforming Fifa a "key priority" as pressure from governments around the world for a fundamental overhaul of world football's governing body grew in the wake of fresh allegations of bribery and corruption during the World Cup bidding process.

As Fifa promised to investigate claims from the former Football Association chairman Lord Triesman that four Fifa executive committee members asked for money or honours in return for their vote, and further claims made in the Sunday Times that two more accepted bribes of $1.5m (£900,000) from Qatar, the Conservative sports minister insisted pressure on the organisation would increase significantly.

Hugh Robertson said an international consensus was forming that Fifa should be made to reform in the way the International Olympic Committee was forced to change after the Salt Lake City scandal in 1999. Then, 10 IOC members were expelled or forced to resign over allegations of vote-buying during Salt Lake City's winning bid for the 2002 Winter Olympics.

"I would like to see really concerted pressure from international bodies to get them to reform," said Robertson. "We're pretty wound up about it, the Dutch [and] the Australians are pretty wound up and there are a number of others."

He said finding a way to force Fifa to reform would be one of the "key objectives" for Poland, which takes over the running of the EU Council from July. "Cleaning up and reforming international sports institutions is a key objective of the Polish presidency for next year," he said. "If the commission take an interest, if whatever country is holding the presidency takes an interest, if we can sustain that for more than one cycle, then we have a chance. No organisation likes being held up to international ridicule and constantly being told they are corrupt. It's got to be much more transparent. They have got to be much more open and much more transparent."

Sepp Blatter, who is increasingly likely to win a fourth term as Fifa president on 1 June, and his challenger Mohamed Bin Hammam have promised reforms to Fifa's structure and procedures, including the World Cup bidding process. But there remains widespread cynicism about whether they will deliver.

Blatter said the allegations would be dealt with before the body's congress in Zurich in three weeks' time, when the 73-year-old hopes to win another four years as president. "We have to do it very fast," he told al-Jazeera. "We have a Congress to come and have to deal with this matter before the Congress and not just kick it out of the minds of Fifa and [say] we will deal with it afterwards."

"We have to do it now, immediately, and we have three weeks.We must accelerate the movement, whether it is for the good or for the bad."

All four of the executive committee members accused by Triesman have denied wrongdoing. Brazil's long-standing federation chief, Ricardo Teixeira, described as "absurd" the allegation that he asked Triesman to "come and tell me what you have got for me". Teixeira said he would pursue all possible "legal action against Triesman", although it is unclear how he would do so under English law.

As Fifa demanded evidence relating to the claims made by Triesman under parliamentary privilege, the FA said its general secretary, Alex Horne, had written to the world governing body offering its full assistance.

Fifa said in a statement that its general secretary, Jérôme Valcke, had written to the FA and expressed "extreme concern" at the allegations "questioning the integrity of some Fifa ExCo members in connection with the bidding procedure for the 2018 and 2022 Fifa World Cups".

Bin Hammam, the Qatari president of the Asian Football Confederation who played a key role in securing the 2022 World Cup for his country, denied bribes were paid to Issa Hayatou and Jacques Anouma. "I can assure you nothing like this has happened from our side. If someone wants to damage reputations like this then they have to provide the proof. You can't just accuse people just like that. It didn't happen. It is fine to say something, to try to damage the reputation of somebody but where is the proof?"

In addition to the allegation involving Teixeira, Triesman claimed the Fifa vice-president Jack Warner asked for cash to build an education centre and buy World Cup TV rights for the people of Haiti; that Thailand's Worawi Makudi wanted to be given the TV rights to a friendly between England and Thailand; and that Paraguay's Nicolás Leoz asked for a knighthood.

Warner said he "laughed like hell" at Triesman's claims: "First of all, I laugh like hell because it took those guys from December to now [to say] that I have £2.5m, I believe. I never asked anybody for anything. When these guys came here, we promised to help. I showed them a place where they can put a playground. They promised to come back but they never did. That's all."

Leoz's spokesman called the accusations "pure fantasy and morbid", and a statement issued on behalf of Hayatou, the head of African football, said "he has categorically denied allegations of corruption brought against him before parliament in Britain. "This kind of reporting to create and propagate false information to destroy his reputation, leadership and integrity will not succeed. The president of CAF said all these accusations brought against him are pure invention and an attempt to discredit him."