James Dingemans QC will examine the evidence and speak to Triesman, who made the allegations under Parliamentary privilege at the House of Commons on Tuesday, during an inquiry into the reasons for the bid's failure.
"(Lord) Triesman has made accusations about people involved in the process which are very serious. It is essential that we determine as soon as possible the weight of evidence behind these serious allegations," current FA chairman David Bernstein told reporters on Thursday.
Dingemans has until May 27 to report, when his findings will be made public.
Bernstein added that the FA originally had no plans to mount an inquiry into their failed bid to secure the 2018 finals but had changed its mind in the light of Triesman's allegations.
The world governing body FIFA has since contacted the FA to ask for all possible evidence which it said should be with them four days before its Congress begins in Zurich on May 31.
Triesman alleged that four FIFA members personally asked him for favours during the World Cup bid campaign.
He said FIFA vice-president Jack Warner asked for 2.5 million dollars to pay for an education centre in Trinidad and a further 500,000 for TV rights for Haiti, which he owned.
Triesman said Paraguayan Nicolas Leoz, the president of the South American confederation, had asked for a knighthood.
Thailand's Worawi Makudi demanded television rights to an England friendly with Thailand while Brazil's Ricardo Teixeira said "you come and tell me what you have for me", Triesman added.
All four have denied the claims.
Meanwhile, Bernstein confirmed that the FA would debate whether to back the incumbent Sepp Blatter or his Qatari challenger Mohamed Bin Hammam who go head-to-head in the FIFA presidential election at the FIFA Congress in Zurich on June 1.
"The key thing is openness in process, in financial information, in the election procedures," he said.
"FIFA is a very closed organisation and a lot goes on behind closed doors. We would like to see those doors open much more. "
