Triesman was giving evidence on Tuesday to a British parliamentary inquiry into the reasons why England failed to secure the finals which were awarded to Russia last December.
Members of Parliament involved in the inquiry also revealed the names of two other FIFA Executive Committee members who, it is alleged, were paid $1.5 million to vote for Qatar's successful 2022 World Cup bid.
Conservative MP Damian Collins said the committee had evidence from the Sunday Times newspaper which it would publish that FIFA vice-president Issa Hayatou of Cameroon and Jacques Anouma of the Ivory Coast were paid by Qatar.
Two other executive committee members were banned by FIFA's Ethics Committee last year after a previous Sunday Times investigation into the World Cup bidding process.
The claims are an embarrassment for the game's governing body with a total eight of its 24-strong key decision-making executive committee having now been accused by the British media or its parliamentary representatives of corruption.
Its 75-year-old president Sepp Blatter will stand for a further four-year term at its helm on June 1 in Zurich. He was first elected in 1998. Asian Football Confederation chief Mohamed Bin Hammam is opposing him.
Triesman spoke at the parliamentary inquiry of the "improper and unethical behaviour" by the four men he named.
He said the Concacaf's Warner asked for 2.5 million pounds to be "channelled through me" for an education centre in his home country Trinidad and Trinidad.
HAITI EARTHQUAKE
After the Haiti earthquake struck leaving that country devastated, Warner then asked Triesman for 500,000 pounds to buy Haiti World Cup TV rights.
Triesman said Paraguayan Leoz, president of South America's Conmebol, had requested a knighthood in return for his vote while Brazil's CBF chief Teixeira told him "Come and tell me what you have for me."
Thai Makudi wanted control of the television rights for a proposed Thailand v England friendly.
"We had a number of conversations with Mr Makudi, telephone conversations," Triesman said.
"These were some of the things that were put to me personally, sometimes in the presence of others, which in my view did not represent proper and ethical behaviour on the part of members of the executive committee," he added.
Leoz declined to comment on Triesman's claims but the CBF issued a statement in which Teixeira said he was starting legal proceedings against Triesman.
Teixeira said an English delegation had asked him for his vote in an April 2010 meeting at CBF headquarters but Triesman was not present and he told them Brazil had joined in South American support for the joint Spain/Portugal bid.
"The president of the CBF is already taking the relevant judicial measures with a case against Mr David Triesman for the absurd comments which in truth attempt to hide his failure in leading the English candidacy since it onl
