A Sunday Times report claims that former Asian Football Confederation chief Mohammed bin Hammam paid up to €3.7 million (£3m) to football officials in order to gain support for Qatar's bid.
Qatar have always denied the former Fifa vice-president, who was banned from all football activity for life in July 2013 amid allegations of bribery, worked on their behalf in the run-up to the vote. He was unavailable for comment when the newspaper attempted to contact him.
Qatar were awarded the tournament in December 2010, beating off competition from the United States, South Korea, Japan and Australia.
However, the bid has since been shrouded in controversy, with the searing Qatari summer heat expected to see the event moved to the winter months between November and January, while the country's human rights record has come under immense scrutiny in the British press.
Several figures in world football's governing body, including president Sepp Blatter, have admitted fault in overlooking the climate in Qatar as a potential problem.
And Boyce, speaking on Sunday morning, has welcomed an investigation into the latest claims, putting faith in Fifa's chairman of the investigatory chamber of the Ethics Committee, former United States Attorney, Michael Garcia.
Boyce told BBC Radio 5 Live: "As a member currently of the Fifa executive committee, we feel that any evidence whatsoever that people involved were bribed to do a certain vote, all that evidence should go to Michael Garcia, whom Fifa have given full authority to, and let's await the report that comes back from Garcia.
"If Garcia's report comes up and his recommendations are that wrongdoing happened for that vote for the 2022 World Cup, I certainly as a member of the executive co would have absolutely no problem whatsoever if the recommendation was for a re-vote.
"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 at that time, there's no doubt about that."