Qatar's controversial success in taking the 2022 World Cup to the desert was propelled by millions of dollars in bribes, according to previously unpublished conversations key figures connected with Fifa held with undercover reporters.

In evidence published on Tuesday under parliamentary privilege by the select committee on football governance, the Sunday Times alleges that Michel Zen-Ruffinen, a former secretary general of Fifa, introduced the reporters to a certain Amadou Diallo. Zen-Ruffinen is said to have claimed that Qatar was "using Diallo to arrange financial deals with the African [Fifa executive committee] members in exchange for World Cup votes".

Diallo is said to have been a senior staff member in the entourage of Issa Hayatou, the Confederation of African Football's president. Ismail Bhamjee, himself a former member of the Fifa executive committee, was allegedly more explicit in his dialogue with the undercover reporters.

According to the letter sent by the Sunday Times to John Whittingdale, the chairman of the select committee inquiry into football governance, Bhamjee explained that some of Africa's current and former representatives on the Fifa executive committee had previously sold their votes. "Bhamjee said … Issa Hayatou of Cameroon, Slim Aloulou of Tunisia and Amadou Diakite of the Ivory Coast had each been paid for their votes by Morocco when it was bidding against South Africa in the contest for the 2010 World Cup."

Bhamjee is directly quoted in the letter as saying: "I'm told the Africans will get … anything from a quarter to half a million dollars," in reference to alleged payments from the Qatar bid to African executive committee members. Asked to clarify whether that money was to invest in football or for them personally, Bhamjee reportedly replied: "No, no, no, no. This is on top. This is separate from the football."

Damian Collins, a Conservative MP on the select committee, put these allegations to Mike Lee, a strategic communications consultant to the Qatar 2022 bid who was appearing as a witness. Lee responded: "I was working at the highest level of that bid and talking at length with the chairman and CEO and saw no evidence of any of these allegations.

"My experience is I would have had a sense if such things were going on and I had no sense of that."

On Tuesday night the Qatar Football Association issued a statement in which they said they "categorically deny" the allegations. "As the Sunday Times itself states, these accusations 'were and remain unproven'. They will remain unproven, because they are false," it said.

Specifically the Sunday Times claims that Diakite, the former Fifa executive committee member who remained intimately involved in the Fifa machine, talked about $1m (£611,000) to $1.2m payments for "projects by Qatar in return for their 2022 vote".

The newspaper says it highlighted these several allegations to Fifa in its own submission to the governing body. It comments: "Fifa does not appear to have pursued any of these matters."

A London law firm hired by the Qatar bid has strongly denied the allegations, calling them "entirely false". However the Sunday Times's letter said that it had separately spoken to "a whistleblower who had worked with the Qatar bid". That individual is said further to have claimed that Hayatou and his fellow executive committee member from the Ivory Coast, Jacques Anouma, had received $1.5m from Qatar "to secure their votes".

It added: "The cash was to go to the three members' football federations but there would be no questions asked about how the money was used." It quoted the whistleblower as saying: "Basically if they took it into their pocket we don't give a jack."