Meanwhile, Coe - along with Thompson, 2018 chief executive Andy Anson, 2012 official Sir Keith Mills and other board members and bid executives - will continue talking to Fifa to explain the circumstances behind Triesman's departure.
Double Olympic gold medallist Coe took leave of absence from his position as chairman of Fifa's ethics committee in February 2009 after he was appointed as a non-executive member of the bid board.
Mills, the deputy chairman of the London Organising Committee and a Tottenham director, stepped down from the 2018 bid board when it was streamlined late last year.
Russian bid chief Alexey Sorokin responded by describing the allegations as "absurd" and urged Fifa to "take appropriate measures" ahead of the decision on who will host the showpiece event, which is expected to be made in December.
England fans 'fed up' with press
But former England keeper Peter Shilton believes there will be no long-term impact on the World Cup bid.
"One person does not make a bid," said Shilton, who holds a record 125 England appearances and is an ambassador for the 2018 bid.
"Everybody makes mistakes, it has happened in the past and will happen in the future.
"I am part of the World Cup bid, and feel it does not take anything away.
"Things do die down, and it is the actual substance of the case, which I think is very strong for us to host the World Cup, is what will be the deciding factor."
Triesman was secretly recorded allegedly divulging sensitive information to a former aide, including a claim that Spain and Russia, rival bidders for the 2018 World Cup, were conspiring to bribe referees at next month's finals as part of efforts to win the right to host the tournament.
The former Labour peer accused the Mail on Sunday newspaper of engaging in "entrapment" tactics in order to cause him personal embarrassment.
Lord Triesman resigns as FA chairman
"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."
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 them the event for the first time since 1966.
Triesman has also resigned as FA chairman and will be replaced by board members David Sheepshanks and Roger Burden who will be drafted in as acting joint-chairmen.
The revelations came only two days after the FA submitted their 1,752-page bid book to try to persuade Fifa to award England the 2018 event.
2018 BID TEAM BOARD
Geoff Thompson - chairman
Andy Anson - chief executive
Lord Sebastian Coe
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.
And the new the minister for sport and the Olympics, Hugh Robertson, is certain England's bid team can recover.
"I don't think anybody would pretend it's been particularly helpful," Robertson told BBC Radio 5 live.
"But London 2012 went through exactly this process. You may remember there was an extraordinarily unhelpful Panorama programme that attacked the entire integrity of the IOC.
"After that, London 2012 were told that their bid was dead but they overcame that.
"Whatever Lord Triesman may or may not have said, he didn't attack Sepp Blatter personally, or the entire integrity of either Fifa or Uefa.
"In some ways this is an offence of a much lesser order."
As for the validity of the reported claims from Triesman, Robertson said: "Nothing I have heard at any stage in the process, either in opposition or in Government, suggests there is any truth in these allegations whatsoever."
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