The Fifa presidential election will take place on Friday as planned despite the arrest of seven Fifa officials on charges they received more than $150m (£100m) in bribes.
Among those arrested is Fifa vice-president Jeffrey Webb.
A separate criminal investigation into how the 2018 and 2022 World Cups were allocated has also begun.
But Fifa has already ruled out a revote, insisting Russia and Qatar will remain the respective hosts.
Sepp Blatter faces Prince Ali bin al-Hussein in Friday's election as he seeks a fifth term as Fifa president.
Prince Ali described Wednesday's developments as "a sad day for football" but declined to comment further.
Fifa issued a statement welcoming "actions that can help contribute to rooting out any wrongdoing in football".
Corruption probe number one
Swiss police made dawn raids at the Baur au Lac hotel in Zurich, where Fifa officials are staying ahead of Friday's presidential election.
The seven Fifa officials were arrested after the US Department of Justice issued a 47-count indictment charging 14 defendants with racketeering, wire fraud and money laundering conspiracies in a 24-year scheme.
The Federal Bureau of Investigation, a US law enforcement organisation, also raided the headquarters of the Confederation of North, Central America and Caribbean Association Football (Concacaf) in Miami.
"The indictment alleges corruption that is rampant, systemic and deep-rooted," said US attorney general Loretta Lynch.
"It spans at least two generations of soccer officials who, as alleged, have abused their positions of trust to acquire millions of dollars in bribes and kickbacks."
- Bribes linked to commercial deals dating back to the 1990s for football tournaments in the US and Latin America
- Nine current or former Fifa officials, including Webb and former Fifa vice-president Jack Warner, charged with corruption
- Four individual and two corporate defendants, including former Concacaf general secretary Chuck Blazer and Warner's sons Daryan and Daryll, plead guilty
- Alleged scheme "fostered a culture of corruption and greed that created uneven playing field"
- "Undisclosed and illegal payments, kickbacks and bribes became a way of doing business at Fifa"
- Majority of scheme involved corruption over media and marketing rights to matches and tournaments
The seven Fifa officials arrested were Webb, Eduardo Li, Julio Rocha, Costas Takkas, Eugenio Figueredo, Rafael Esquivel and Jose Maria Marin. They now face extradition to the United States.
Six of the seven are challenging their extradition to the US, Swiss officials later revealed.
The two indicted defendants who were not among the seven arrested in Zurich are Warner and former Fifa member Nicolas Leoz.
Warner, a former Fifa vice-president from Trinidad and Tobago, insists he is innocent of the charges.
Now 72, he quit world football's governing body in 2011 after being suspended pending the outcome on an investigation into corruption allegations against him.
In a statement, he said the "actions of Fifa no longer concern me".
Corruption probe number two
In a separate move, just hours after the earlier arrests, Swiss authorities opened criminal proceedings over the awarding of the hosting rights to the 2018 and 2022 World Cups.
Electronic data was seized from Fifa's headquarters in Zurich as part of a investigation into "criminal mismanagement" and "money laundering".
Swiss police intend to question 10 Fifa executive committee members who took part in the voting process in December 2010.
Three years ago, Fifa hired a former US prosecutor, Michael Garcia, to investigate allegations of bribery over the awarding of the World Cups to Qatar and Russia.
However, it refused to publish his report, releasing only a summary in which it said there were no major irregularities. An angry Garcia quit, saying the summary was "erroneous".
Reacting to the news that criminal proceedings had been opened, Russian sports minister Vitaly Mutko said his country had "nothing to hide" and welcomed the Swiss investigation.
Fifa's reaction to developments
World football's governing body has been dogged by allegations of corruption and wrongdoing in recent years.
The latest events are another major blow, although Fifa's communications director insisted the organisation was committed to reform.
"This is good for Fifa," Walter De Gregorio told a hastily arranged news conference on Wednesday. "It hurts, it is not easy, but it confirms we are on the right track."
De Gregorio said Blatter was "very calm", "is fully co-operating" but added that the 79-year-old Swiss "is not dancing in his office".
Fifa later issued a statement welcoming "actions that can help contribute to rooting out any wrongdoing in football".
The rest of the world reacts
Uefa, the body that governs European football, said it was "astonished and saddened by the events", while Jim Boyce, Britain's outgoing Fifa vice-president, said it was "another sad day for Fifa".
German Football Association president Wolfgang Niersbach said developments were "shocking and harmful for the world of football".
He added: "It would be terrible if the severe accusations against members of Fifa prove to be correct."
England's Football Association said Wednesday's events "are very serious for Fifa and its current leadership".
FA chairman Greg Dyke added that there "must be a question mark" over whether the presidential election should take place now, while former chairman David Bernstein has said it should be postponed.
Damian Collins, a British member of parliament who has called for Fifa reform, said Blatter had failed to live up to his promises to stamp out corruption.
"Because he has totally failed to do this, it has been left to an outside law enforcement agency to do the job and take action," said Collins.
Former England captain Gary Lineker tweeted: "There can't be a more corrupt, deplorable organisation on earth than Fifa. The house of cards is falling. Time for change!"
He added: "This is extraordinary! Fifa is imploding. The best thing that could possibly happen to the beautiful game."
Fifa sponsor Adidas also reacted to developments: "Following today's news, we can only encourage Fifa to continue to establish and follow transparent compliance standards."
The German sportswear company, associated with Fifa since the 1950s, had already expressed concern about allegations of corruption connected to the awarding of the 2022 World Cup to Qatar.