The FA's acting chief executive Alex Horne responded: "It's important that we can demonstrate to Fifa and the rest of the world at this time that we are serious about our bid for World Cup 2018. The gossip and the nonsense doesn't matter."
Triesman also quit his role as FA chairman after he was secretly recorded allegedly divulging sensitive information to a former aide.
The allegations included a claim that Spain and Russia, rival bidders for the 2018 World Cup, were conspiring to bribe referees at next month's finals in South Africa as part of efforts to win the right to host the tournament.
The former Labour peer, who also resigned from his post as chairman of the FA, accused the Mail on Sunday newspaper of engaging in "entrapment" tactics in order to cause him personal embarrassment.
"In that conversation I commentated on speculation circulating about conspiracies around the world," said Triesman.
"Those comments were never intended to be taken seriously as indeed is the case with many private conversations."
The FA has apologised for Triesman's claims to the Spanish and Russian governing bodies, with copies of the letter obtained by a number of news agencies.
"England 2018 unreservedly apologises for these comments, for any suggestions of any improper behaviour on the part of any members of the Russian football family, our fellow bidders and for any express or implied criticism of the Russian Football Federation or of Fifa," part of the letter read.
"The comments reported to have been made by Lord Triesman in no way represent the views of England 2018, any employee, director, ambassador, consultant or adviser to the bid.
Damage to World Cup bid 'not fatal'
"England 2018 bitterly regrets any damage to the integrity of the Russian football family, Fifa or any of its member associations caused by these comments. We are available to discuss this matter at your convenience."
Before Triesman's unwanted spell in the headlines, England's 2018 World Cup candidacy was generally viewed as having a good chance of success.
But the bid team now faces an uphill task to persuade Fifa's executive to award England the event for the first time since 1966.
The revelations came only two days after the FA delegation submitted its 1,752-page bid book to Fifa.
A European bid is tipped to get the 2018 tournament with England up against Russia and joint bids from Spain/Portugal and Belgium/Netherlands.
The other bidders, although they are mainly focused on the 2022 tournament, are Australia, the United States, Japan, Qatar and South Korea.
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