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 following a series of arrests, including seven of its officials in Zurich. The arrests came as part of an FBI investigation and separate probes by Swiss authorities into the bidding for the 2018 and 2022 World Cups.
There were also revelations of bribes paid for votes for the 2010 World Cup in South Africa and increasing concerns from FIFA’s sponsors.
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 pay 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.”
In stressing a determination that sponsors such as Coca-Cola, Visa and McDonalds will not escape scrutiny, they state that “the whole of FIFA entourage needs examining”.
Amid the growing public anger being aimed at sponsors, Mr Norman and Mr Collins write: “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 the other sponsors to put their weight behind this as well.”
They called for a full independent reform process for FIFA, stating that only “a respected international figure” such as former United Nations secretary general Kofi Annan should be put in charge of it. It must also co-operate with all the ongoing legal investigations, they suggest.
The arrests came as part of a US Department for Justice investigation which described the allegations as “corruption that is rampant, systemic and deep-rooted”.
Former FIFA vice-president Jeffrey Webb from the Cayman Islands has agreed to be extradited to the United States from Switzerland.
He appeared in court in New York on Saturday and pleaded not guilty to charges including bribery and fraud. He was released on bail after his family posted a $10 million bond, but 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 will come under pressure on Monday to agree to an early date for the election of his successor at the first meeting of the world governing body’s executive committee since the corruption crisis exploded.
The meeting in Zurich will allocate the date for a special congress to elect a new president after Mr Blatter’s announcement last month that he will step down.
Those pushing for Mr 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.
Mr Platini is among the favourites to stand for the FIFA presidency, and were he to win then Mr Niersbach would be a likely successor to the Frenchman for the UEFA post.