The England 2018 team believe African confederation president Issa Hayatou voted for them, along with bid chairman Geoff Thompson.
A senior government source has told the BBC that at least five executive committee members "personally assured" Prince William and David Beckham they would vote for the England bid, only to back other contenders in the first round.
The source said Concacaf president Jack Warner and general secretary Chuck Blazer were among the executives who had given the assurances.
Russia (nine votes), Spain-Portugal (seven votes) and Netherlands-Belgium (four votes) went through to a second round of voting.
Russia then earned an absolute majority of 13 votes, with Spain-Portugal on seven and Netherlands-Belgium picking up two.
And despite suffering a number of setbacks during the bidding campaign, there was a feeling that England had made great strides in the build-up to the vote.
First round: England 2 votes (eliminated), Netherlands-Belgium 4; Spain-Portugal 7; Russia 9. (No absolute majority)
Second round: Netherlands-Belgium 2; Spain-Portugal 7; Russia 13. (Russia obtain absolute majority)
Prince William, Prime Minister David Cameron and Beckham lobbied hard for support, but Anson suggested executive committee members went back on their word by backing for England's rivals.
"I still find it hard to understand what happened," he told a media conference in Zurich on Friday morning.
"When you have the best technical bid, fantastic inspection visits, the best economic report, and, from what people told us, the best presentation, it's quite hard to stomach that all that seemed to count for absolutely nothing.
"I'm not going to beat around the bush - individual members promised to vote for us and didn't clearly.
"Russia did a lot of last-minute lobbying and votes appeared to switch at the last minute - we know some switched in the early hours of the morning."
Anson was repeatedly pressed on who did not deliver on their promise of voting for England, but he refused to reveal any names.
First round: Australia 1 (eliminated); Japan 3; United States 3; South Korea 4; Qatar 11. (No absolute majority)
Second round: Japan 2 (eliminated); South Korea 5; United States 5; Qatar 10. (No absolute majority)
Third round: South Korea 5 (eliminated); United States 6; Qatar 11. (No absolute majority)
Fourth round: United States 8; Qatar 14. (Qatar obtain absolute majority)
But he was furious with the voting process and called for extensive changes to be implemented.
Only 22 of Fifa's 208 national football associations were included in the ballot, and the voting pattern of individual delegates was kept secret.
"You have got to open it up to all the member associations. You've got to widen the electorate," stated Anson.
"For me, you should have transparency and open voting so that everyone knows who voted for whom, because I don't believe that the secret ballot actually helps transparency at all.
"It leads to the situation we had on Thursday where people promise you something and don't deliver."
Anson added that the England 2018 team went to bed on Wednesday feeling the bid had "room to manoeuvre" in terms of votes, and confident they would get through the first round.
"To then find you only get two votes is tough and I'm still finding that hard to believe," he said.
While Russia were awarded the 2018 competition, there was a huge shock in the 2022 ballot when Qatar prevailed with an absolute majority of 14 votes.
Australia had a very good bid and got one vote, we had a very good bid and got two, the USA had an unbelievably strong technical bid and got three. Six votes in the first round between those three, there's something not quite right
Australia received the best Fifa evaluation report but were eliminated in the first round with only one vote as Qatar picked up 11, Korea four, the United States three and Japan three.
Japan fell in the second round with two votes after Qatar picked up 10, Korea five and the US five.
In round three, Korea went out with six votes as Qatar (11 votes) and the US (six vote) went on to the final stage.
"Running two bids together was clearly a huge mistake," insisted Anson. "Everyone who had a vote and a bid clearly wanted to trade that vote for something that helped them get over the line in that campaign.
"Australia had a very good bid and they got one vote, we had a very good bid and we got two, the USA had an unbelievably strong technical bid and got three.
"Six votes in the first round between those three, there's something not quite right."
Ron Walker, a member of the Australian Football Federation, described the current voting system as a disgrace.
Fifa has outgrown its roots - Mark Palios
"There has been some contamination of the voting system in Fifa in the last 12 months," fumed Walker.
"There's been a lot of publicity about the way the voting system works and I believe it's time it's changed, otherwise you won't get countries like Australia, England and the US bidding because they know very well the voting system is flawed.
"It's a disgrace to think countries can spend so much money and be encouraged to put forward their best bid when the decision is made some days before."
Former England manager Graham Taylor suggested it was about time the organisation of Fifa was investigated.
"You have got in Fifa an organisation that doesn't have to answer to anyone, even governments," he told BBC Radio 5 live. "If governments get involved, Fifa suspend the national team.
"It might now be time they really need looking into. They should really be investigated, and of course our journalists are very good at that."
It is the biggest game in the world, and they have got to say, 'Are we fit for purpose for the 20th century?' I think the answer to that is no
Former sports minister Richard Caborn
Former Football Association chief executive Mark Palios urged Fifa to replicate the structure of the International Olympic Committee, with a wider electorate.
"There is a very simple way to change it and that is to expand the size of the constituency that votes from 22 - or 24 - people to roughly the size of the Olympics situation where it is over 100," Palios explained.
"If you wanted to go for real democracy, why can't everybody within Fifa to have a vote. That would then make it easier for genuinely the best bid to be voted on by the football family."
Former sports minister Richard Caborn questioned whether Fifa was fit for purpose, adding: "I think there has got to be more transparency, there has got to be more democratic accountability.
"All that needs to be looked at because it only brings itself into disrepute if it does not do that.
"It is the biggest game in the world, and they have got to say, 'Are we fit for purpose for the 20th century?' I think the answer to that is no."
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