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.
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.
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.
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."
"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.
"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
