The Serious Fraud Office is looking into claims of corruption in the bidding process for the 2018 and 2022 World Cups and is seeking to gather evidence for a possible criminal case.
Conservative MP Damian Collins has been pushing for an investigation amid persistent allegations of corruption and bribery relating to the decision to award the tournaments to Russia and Qatar respectively.
The SFO has written to Mr Collins to state that "...every reasonable line of inquiry, including working closely with appropriate overseas authorities, is being pursued to ascertain whether the director has grounds to open an investigation".
FIFA recently called in the Swiss authorities to consider whether criminal proceedings should be instigated against individuals involved in the bid process.
Earlier this month, the release of Judge Hans-Joachim Eckert's summary of Michael Garcia's report into the matter descended into confusion as Garcia distanced himself from its conclusions. FIFA has so far resisted repeated calls for the report to be published in full.
"We know that FIFA have turned over their own internal investigation to the Swiss authorities and the FBI are already investigating: I think there are grounds for the SFO to follow suit," Mr Collins told Sky Sports News HQ.
"I want the force of international law to come down on FIFA. They seem to regard all these issues as private matters for them but serious allegations of bribery and corruption are rightly matters for law-enforcement agencies."
I want the force of international law to come down on FIFA. They seem to regard all these issues as private matters for them.
The SFO had previously indicated that any alleged wrongdoing would not fall within its jurisdiction, but Mr Collins said the latest developments in the fast-moving story had changed things.
"Last week we saw significant movement even by FIFA," he said.
"We know from the summary report they published that the Garcia report contains evidence from UK nationals and we know that UK nationals worked on the bid teams of other countries and we also know from FIFA's own words that the report and its documents contain what they regard as evidence of unlawful activities.
"So there are very real grounds for the SFO to believe they should investigate this and there could be grounds for their own criminal investigation. Clearly by following this up with other agencies they're looking to establish what sort of investigation they could launch."
Mr Collins said he believed the football fraternity had lost faith in FIFA's attempts to keep the investigations in-house.
I don't believe Sepp Blatter will ever willingly give up this information that he's protecting.
"I think the reaction to events of the last couple of weeks has been that people have been horrified by the way FIFA have sought to deal with these serious allegations," he said.
"They give the impression of wanting to hide them away. It's external action that will lead to change.
"I don't believe Sepp Blatter will ever willingly give up this information that he's protecting. But the FBI and the SFO and so on have the force of international law to go after it and I hope that they do that.
"They've failed time and time again to take these allegations seriously, to get to the bottom of the wide-ranging allegations that have dogged the bidding process. They've failed every time and I'm afraid that under the current leadership they will fail again.
"It requires new leaders in FIFA or maybe the chance to completely restructure and reform the organisation. It will be that level of reform that will be required to put FIFA on a proper footing and make it into a body that can run the world game in a way that fans can respect."