Claims made in the Sunday Times have put the award of the 2022 World Cup to Qatar in the spotlight, although organisers vehemently deny any wrongdoing.
Triesman spearheaded England’s bid for the 2018 World Cup before stepping down from his role as FA chairman in May 2010, seven months prior to the voting for the 2018 and 2022 tournaments.
The 70-year-old admits it would not surprise him if rules were broken during the bidding for the tournaments and insists there is a strong argument to restart the entire process should evidence of wrongdoing emerge.
“There is a very strong argument for that (restarting the bid process for both tournaments),” Triesman told Sky Sports News.
“Everybody said there was to be no trading of votes between the two competitions and I always said if you run the two competitions together you will always get trading of votes.
“Of course it would be hard to prove. It would be done in private rooms but a lot of the decisions that people took in voting for Russia I have absolutely no doubt were closely connected in the decisions they took in voting for Qatar.
“So in those circumstances I think there is a very good case for opening the issues up again. I don’t think you can leave the World Cup in any location where it has been awarded by means that are not legitimate. It would be tainted.
“If it is found the decision was driven by widespread acts of very considerable corruption, and if you live in a society where you think the rule of law has got some importance in the way that you conduct your life, you’ve got to say we can’t live with that decision.”
Earlier this week, FIFA president Sepp Blatter appeared to suggest the allegations published by the Sunday Times were part of a campaign based on discrimination and racism orchestrated by the British media.
Lord Triesman condemned Blatter’s remarks, adding he believes the 78-year-old should be replaced as the head of world football’s governing body to protect the reputation and integrity of the game.
“I think the idea that the press here are motivated by racism is a grotesque allegation,” Triesman said.
“The reality is that there has been huge amounts of work done on racism in English football usually by the same people who have been most concerned by corruption in football and I just think it’s a way of trying to curry favour in an election.
“It’s an electoral stunt and he really ought to be given no latitude by us in saying that it’s grotesque, it’s untrue and we have a press that is not motivated that way.
“When you look at it his presidency and the presidency before it (of Joao Havelange), there have been periods in which there has been massive controversy. Terrible, terrible, allegations that were never really investigated properly.
“He has been the head of a culture which I think has been disastrous for world football. You have to change that culture and the best way to change that culture is to change the leadership.”
Earlier in the day,Triesman used parliamentary privilege to launch a stinging attack on FIFA during a debate in the House of Lords.
"FIFA, I'm afraid, behaves like a mafia family. It has a decades-long tradition of bribes, bungs and corruption," he said.
"About half of its executive committee who voted on the last World Cup have had to go.
"Even its past president Joao Havelange has been removed from his honorary life presidency in his 90s.
"Systematic corruption underpinned by non-existent investigations where most of the accused are exempt from the investigation make it impossible to proceed.
"Foreign construction workers dying in their dozens in Qatar stadium construction sites are essentially ignored."