Fifa investigator and New York lawyer Michael Garcia plans to complete his probe of the 2018 and 2022 World Cup bidding process by 9 June.
New allegations of corruption surrounding the 2022 tournament in Qatar surfaced on Sunday.
Garcia met Qatari officials on Monday and said he had spent "months" interviewing and gathering materials.
He said he planned to submit a report of his findings to Fifa's adjudicatory chamber towards the end of July.
"The report will consider all evidence potentially related to the bidding process, including evidence collected from prior investigations," said Garcia in a statement.
The Sunday Times reported a number of football officials took £3m in return for support of the Qatari bid.
The head of the Asian Football Confederation (AFC) has expressed "grave concerns" over the allegations.
Qatar's 2022 bid committee denies "all allegations of wrongdoing".
The results of the 2018 and 2022 World Cup bids were announced at the same time, with Russia winning the vote to stage the 2018 event.
In a statement, AFC president Shaikh Salman bin Ebrahim Al Khalifa expressed his concerns about the reports of corruption but said he was "convinced" the Qatar committee would do "their utmost to clear the doubts".
Shaikh Salman is also leading Fifa's consultation over whether the Qatar finals need to be controversially moved to the winter, rather than its customary spot in June and July.
Former attorney general Lord Goldsmith, a member of Fifa's independent committee on governance, said there should be a re-run of the 2022 vote if allegations of wrongdoing prove correct.
"If it is proved that the decision to give Qatar the World Cup was procured by - frankly one can describe it no other way - bribery and improper influence, then that decision ought not to stand," he told BBC Radio 4.
Fifa vice-president Jim Boyce also said he would support a re-vote if corruption allegations can be proven.
Boyce told BBC Radio 5 live's Sportsweek programme: "If Garcia comes up with concrete evidence - and concrete evidence is given to the executive committee and to Fifa - then it has to be looked at very seriously."
Mark Pieth, a law professor appointed by Fifa president Sepp Blatter to lead the independent committee on governance, called the latest revelations "exciting".
In an interview with BBC World Service's Newsday programme, he also said he hoped the Sunday Times would share its information with Garcia.
Football Association chairman Greg Dyke has said a new vote should take place if it was shown a "corrupt system" led to Qatar's win, while UK Sports Minister Helen Grant said it was "essential that major sporting events are awarded in an open, fair and transparent manner".
Prime Minister David Cameron said that the inquiry into the bidding process must be allowed to run its course before passing judgement, while Deputy Prime Minister Nick Clegg tweeted that the vote must be re-run if the "shocking allegations" were proven.
However, Mr Cameron added that his experience of Fifa's bidding procedure, when he helped lobby for England to host the 2018 finals, left him unhappy.
"My memories of that bidding process are, as I've said earlier, not happy ones in terms of the way the whole thing was arranged and the role of Fifa and the rest of it," Mr Cameron said.
Allegations of corruption centre on former Fifa official Mohamed bin Hammam.
The Sunday Times claims to have obtained secret documents that implicate the former AFC president in corrupting members of football's governing body to win the right to stage the 2022 World Cup.
The newspaper alleges the documents, seen by BBC sports editor David Bond, show that Qatari Bin Hammam, 65, was lobbying on his country's behalf at least a year before the decision to award the country hosting rights.
They also allegedly show he had made payments into accounts controlled by the presidents of 30 African football associations and accounts controlled by Trinidadian Jack Warner, a former vice-president of Fifa.
A statement from the Confederation of African Football (Caf) said the specific allegations against its president, Issa Hayatou, were "fanciful", "ridiculous" and formed part of a "smear campaign" against him.
It added that Cameroonian Hayatou, a Fifa vice-president, categorically denied the allegations of corruption.
"Mr Hayatou will not allow journalists once again to attack his integrity and reputation," it read. "Such allegations are meant to discredit not only him as a person but the whole continent."
Qatar's bid committee and Bin Hammam have always strenuously denied any wrongdoing.
The committee also rejected claims Bin Hammam actively lobbied on their behalf in the run-up to the vote in December 2010.
It added it was co-operating with Garcia, insisting it will "take whatever steps are necessary to defend the integrity of Qatar's bid".
Qatar defeated bids from South Korea, Japan, Australia and the United States to win the right to stage the 2022 World Cup.
Football Federation Australia (FFA) said it may re-submit its bid to host the 2022 World Cup in the wake of the latest claims.
"It's a serious development, they're serious allegations and we're looking to see what the response will be," said FFA chief executive David Gallop.
He said the FFA had been involved in Fifa's investigation into corruption and the 2010 vote that awarded the World Cup to Qatar.
"We've been heavily involved in this now for many months in terms of the investigation that Mr Garcia is carrying out," added Gallop.
The executive director of Japan's defeated bid, Yuichiro Nakajima, said he would also back a move to re-submit tenders for 2022.
"I can't speak on behalf of the Japan FA - I'm not representing them now - but if I were in the position to say so, yes I would," Nakajima told BBC Radio 4's Today programme.
"The Sunday Times report does point to a number of areas that could be tightened up and sorted out."