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FIFA said the ethics committee's investigatory chamber expects to submit its report, led by prosecutor Michael Garcia, "by the first week of September."
In June, Garcia set a target of late July to deliver his confidential report to ethics judge Joachim Eckert. The investigation file will be used by Eckert to decide on possible sanctions against nine bidders which competed to host the next two tournaments.
Russia and Qatar won hosting rights in December 2010 following votes among FIFA's executive committee. FIFA president Sepp Blatter has said that only his ruling board, and not Eckert, has power to overturn a vote and order a re-run.
Qatar will host the prestigious tournament in eight years' time but the decision has been plagued by allegations of corruption and bribery.
As well as Dyke, the UK Government's Culture Media and Sport (CMS) select committee will also hear from Sunday Times journalists Heidi Blake and Jonathan Calvert.
The newspaper reported in June that Qatari football official Mohamed Bin Hammam made payments totalling three million pounds in return for support for the Qatar bid.
Qatar denies any wrongdoing and claims Bin Hammam always acted independently of the campaign.
Dyke, who was appointed FA chairman in March 2013, has been highly critical of FIFA in the past and has given his backing for a re-vote should the corruption allegations be proven.
The CMS examined allegations of corruption in 2012 and called for a full, transparent investigation. FIFA ethics investigator Michael Garcia is finalising a report into the bidding processes for the World Cup in 2018, won by Russia, and 2022.
Garcia is expected to submit his findings at the end of July with any decision regarding a re-vote to rest with FIFA's executive committee.
UK Prime Minister David Cameron, who was involved in England's 2018 bid, admitted in June that he had misgivings about the campaign.
"My memories of that bidding process are not happy memories in terms of the way the whole thing was arranged and the role of FIFA," Cameron said at the time. "There is an inquiry under way, quite rightly, into what happened in terms of the World Cup bid for 2022.
"We should let that inquiry take place rather than prejudge it."
Despite an unforgiving climate and a lack of suitable infrastructure, Qatar defeated bids from South Korea, Japan, Australia and the United States to win the 2022 vote.
Australia and Japan have already backed calls for the bid to be re-run. The CMS committee will also examine the FA's role and engagement with UEFA and FIFA in the international game.
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