The Mail on Sunday revelations
came only two days after former England captain David Beckham had helped the FA submit a 1,752-page bid book as they try to persuade Fifa to award England the 2018 World Cup.
"I have decided to resign as chairman of the FA and the 2018 Bid board," Triesman said in a statement.
"A private conversation with someone whom I thought to be a friend was taped without my knowledge and passed to a national newspaper," he added, referring to former aide Melissa Jacobs, who met Triesman a fortnight ago.
"That same friend has also chosen to greatly exaggerate the extent of our friendship.
"In that conversation I commentated on speculation circulating about conspiracies around the world. Those comments were never intended to be taken seriously, as indeed is the case with many private conversations.
"Entrapment, especially by a friend, is an unpleasant experience both for my family and me but it leaves me with no alternative but to resign."
Triesman's resignation statement followed an FA board meeting that lasted over two hours.
"It would have been difficult for the FA to have sacked John Terry and for Triesman to have stayed on," said new Sports Minister Hugh Robertson, referring to England coach Fabio Capello's decision to strip Terry of the England captaincy over an extra-marital affair last year.
"I'm very impressed with the way it has been handled," Robertson told the BBC News channel. "The danger was this could have drifted on. I'm pleased they have acted decisively and they have done the right thing.
"Nobody could pretend that this hasn't been a good day. But the trick now is to refocus everyone's attempts on the bid. It can be done and if you concentrate on the bid's core strengths we can get over this."
Apart from the damage to the FA that may be created within the Fifa corridors of power, the world governing body's rules prohibit World Cup bidders from talking about rival bids.
Triesman was quoted in the article as saying: "Spain are looking for help...to bribe the referees".
On Friday, Fifa chief Sepp Blatter spoke in glowing terms of England's 2018 bid which includes 12 towns and cities from Sunderland to Plymouth, calling it "the easiest bid in the world" - but also described the plans put forward by Russia as "remarkable".
After the good publicity that was generated by the 2018 team and Beckham on Friday it remains to be seen what effect Triesman's reported comments are likely to have on the FA's 2018 bid.
From its inception the bid has been troubled by infighting with senior members resigning from the board while Triesman's leadership has also been questioned.
In October the 2018 bid was criticised by Fifa vice-president Jack Warner and Danny Jordaan, who led South Africa's successful 2010 campaign.
A month later former Birmingham City director Karren Brady, who was one of six board members to stand down, said that England's hopes of hosting the 2018 World Cup were in danger of being undermined by internal politics among the bid team.
Beckham delivers 2018 World Cup bid
In November a senior member of Fifa's executive committee returned a handbag given to his wife as a gift by the England bidding team.
The FA tried unsuccessfully to get an injunction against publication of the Triesman story on privacy grounds.
The Mail on Sunday quotes Triesman as saying: "There's some evidence that the Spanish football authorities are trying to identify the referees...and pay them.
"My assumption is that the Latin Americans, although they've not said so, will vote for Spain. And if Spain drop out, because Spain are looking for help from the Russians to help bribe the referees in the World Cup, their votes may then switch to Russia."
But in his statement Triesman said: "The views expressed were not the views of the 2018 bid board or the FA.
"Nobody should be under any misapprehension that the FA or 2018 bid board are disrespectful of other nations or Fifa and I regret any such inference that may have been drawn from what has been reported."
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.
Spain have yet to respond to Triesman's comments, but Russia insisted their bid was committed to "maintaining ethical norms and the principles of fair play".
"I don't know why there are so many rumours regarding Russia's World Cup bid," Russian bid chief Alexei Sorokin told Reuters news agency.
"Maybe because we're moving in the right direction and our rivals see us as a major force and try to derail our bid campaign."
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