Embattled FIFA President Sepp Blatter is facing calls to quit immediately after Swiss authorities launched criminal proceedings against him.
FIFA presidential candidate Prince Ali bin al Hussein says the need for change in the organisation has never been clearer.
And Football Association Chairman Greg Dyke said FIFA was in "meltdown".
On Friday, police opened criminal proceedings against Mr Blatter over corruption allegations.
Mr Blatter is suspected of "criminal mismanagement or misappropriation" and his office has been searched.
The allegations relate to suspicion over a "disloyal payment" of 2m Swiss francs (£1.3m) to UEFA president Michel Platini.
Mr Blatter's American lawyer has denied any wrongdoing on the part of his client.
Prince Ali, the only challenger to Mr Blatter in last May's election, said in a statement that FIFA had been "shaken to its very core by the scandals that have decimated our governing body".
"The need for new leadership that can restore the credibility of FIFA has never been more apparent."
The Jordanian prince said he had spoken to "many member associations" and is confident there is now a hunger for real change in the wake of the latest developments.
"We must now come together and work to restore FIFA's credibility and reputation by bringing about the change that is so clearly needed."
Mr Platini is the favourite to succeed the outgoing Mr Blatter as FIFA president next February, with Prince Ali his main rival.
The payment by Mr Blatter from FIFA funds was "allegedly made for work performed between January 1999 and June 2002".
Mr Platini said the payment "relates to work which I carried out under a contract" and he "was pleased to have been able to clarify all matters" to the authorities. The Frenchman is not accused of wrongdoing.
The opening of the Swiss investigation piled pressure on the 79-year-old Mr Blatter.
FIFA campaigner Damian Collins - a member of the Commons Culture, Media and Sport Committee - told Sky News the international football organisation was at the point of no return and needed to be cleaned out.
"This is an existential crisis for FIFA," the Conservative MP said. "It's clear the corruption allegations spread right across the top of the organisation, right up to Blatter.
"It's impossible to see how they can carry on, how they could lead any kind of reform process or soldier on till February next year when they're supposed to elect their new president.
"Blatter needs to go right now, there needs to be an independent president and an independent reform process that can clean FIFA out and put it back on its feet.
"We are at the point of no return. Sepp Blatter's old FIFA is now dead."
His comments were echoed by Mr Dyke.
"This can't continue," he told Sky News.
"Last week it lost its chief executive and this week the president is being interviewed by the criminal authorities.
"Clearly this is a serious moment in the history of FIFA."