Lord Triesman has stood down as chairman of the Football Association as well as the England 2018 World Cup bid.
His hasty departure follows what he has described as "entrapment" by the Mail on Sunday.
Apprentice star and former Tottenham Hotspur chairman Lord Alan Sugar has voiced interest in replacing Lord Triesman as head of the FA.
Just hours after he presented the FA Cup to Chelsea captain John Terry, the newspaper said that Lord Triesman had suggested Spain could drop its 2018 bid if rival bidder Russia helped bribe referees at this summer's World Cup.
The Mail on Sunday quotes Lord 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."
Lord Triesman: Biography
David Triesman was the son of East End Jewish immigrants, both communists, who taught him to observe International Workers Day as well as Yom Kippur.
He has held positions as the Labour Party's General Secretary, general secretary of the Association of University Teachers, as a Foreign office minister in House of Lords and a parliamentary under-secretary at the Department for Innovation.
He was made life peer after leaving his position with the Labour Party in 2004, becoming Baron Triesman of Tottenham.
It is believed that the the England 2018 team faxed an apology to the Russian and Spanish FAs as it tries to rescue the World Cup bid with a Fifa decision due in December.
A statement from Lord Triesman said: "I have decided to resign as chairman of the FA and the 2018 Bid board.
"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."
Lord Sugar, who no longer has his role as government entreprise tsar since Labour lost the election, told the Sun he was interested in the vacant role. He said: "I would certainly be interested. Not just in terms of the World Cup bid but in terms of reforming the whole FA.
"I have had no contact from the FA but it is an interesting idea and everyone knows I have been a strong commentator on the state of English football. And it's not as if I would be an outsider coming into the game. I do have something of a track record in football."
FA board members David Sheepshanks and Roger Burden have been drafted in as acting joint chairmen of England's governing body, while Geoff Thompson is the new chairman of the World Cup bid.