A parliamentary inquiry wants Fifa to conduct an independent review into the corruption claims that have dogged the organisation, BBC Sport understands.
Football's governing body has been criticised for its response to the 2022 World Cup bribery allegations.
Fifa is also criticised for its handling of corruption claims in their recent presidential elections.
The Culture, Media and Sport select committee's report is due to be published on 5 July.
BBC Sport understands the report will criticise Fifa's "contemptuous" and "dismissive" response to evidence submitted to the committee by the Sunday Times which accused two Fifa members, Issa Hayatou and Jacques Anouma, of asking for bribes to back Qatar's 2022 World Cup bid.
However, the ethics committee decided allegations of vote-trading between Spain and Portugal, who were bidding for the 2018 World Cup, and Qatar, who were bidding for 2022, were unfounded.
The 2018 World Cup was awarded to Russia, while Qatar controversially won the right to host the 2022 tournament.
In May, former FA chairman Lord Triesman accused Fifa executive members Jack Warner, Ricardo Teixeira, Nicolas Leoz and Worawi Makudi of asking for favours in return for their votes, however, Fifa said all four men were cleared of the allegations in an independent report commissioned by the FA.
Later that month, Warner and fellow Fifa member Mohamed Bin Hammam, the Asia confederation president who was due to be Fifa president Sepp Blatter's only challenger in the election on 1 June, were suspended after allegations they paid bribes of £600,000 to Caribbean associations. Both denied the claims.
Bin Hammam withdrew his candidacy ahead of the election, allowing Blatter to be re-elected for a fourth term as president.
In June, Fifa dropped its inquiry into Warner after the former Fifa vice-president resigned from all football activities, but the body will continue its investigation into the conduct of Bin Hammam.
Aside from its criticisms of Fifa, the parliamentary report will also call on the FA to conduct an internal review into the failed 2018 World Cup bid.
England's bid ended in humiliation, earning only two votes and going out in the first round.
Although the FA opened a two-week inquiry into Lord Triesman's claims about Warner, it did not examine the bid as a whole.