Jesse Norman, the new Culture, Media and Sport select committee chairman, declared that “Britain must play its part” in the global drive to clamp down on corruption.
He has ordered an investigation into FIFA reform, due to begin in September, saying that “more can be done”.
FIFA has come under mounting pressure, which has led to a number of arrests of its officials in Zurich as part of an FBI investigation.
The FBI and Swiss authorities have been scrutinising the bidding processes for the 2018 and 2022 World Cups which were awarded to Russia and Qatar.
There have also been revelations regarding bribes been paid for votes which led to South Africa being awarded the 2010 World Cup in South Africa.
Writing in the Daily Telegraph, Mr Norman and select committee member Damian Collins said: “More can be done.
“Alongside the US and Swiss prosecution, other national parliaments and government agencies also have a role to play in using their powers to support FIFA reform. Britain must play its part.
“We will be inviting the FA, the Serious Fraud Office and some of FIFA’s leading global sponsors to explain their actions – or inaction – to date, and to press for urgent reform to the governance of world football.”
Amid the growing public anger being aimed at sponsors, Mr Norman and Mr Collins added: “The question has been raised as to why these corporations have not distanced themselves from FIFA in the face of the charges.
“Coca-Cola has now backed the proposal from the campaign group ‘New FIFA Now’ for an independent reform commission, and it is time for other sponsors to put their weight behind this as well.”
They believe former United Nations secretary general Kofi Annan should be put in charge of such a commission.
As for top officials in the spotlight, former FIFA vice-president Jeffrey Webb from the Cayman Islands appeared in court in New York on Saturday when he pleaded not guilty to charges including bribery and fraud.
He was released on bail after his family posted a $10m bond but he must remain within 20 miles of the court and wear an electronic tag.
He was also ordered to surrender his three passports, two UK passports and one from the Cayman Islands.
FIFA president Sepp Blatter is expected to come under pressure on Monday to agree to an early date for the election of his successor.
The meeting in Zurich will allocate the date for a special congress to elect a new president after Blatter announced last month that he will step down from the post.
Those pushing for Blatter to leave immediately include UEFA president Michel Platini and other European members of FIFA’s executive committee including England’s David Gill and Germany’s Wolfgang Niersbach.