Michael Lauber, the Swiss attorney general, said the "suspicious bank relations" were reported within the framework of anti-money-laundering regulations.
Addressing the media for the first time since the Swiss investigation into FIFA was announced three weeks ago, Lauber said the case is "huge and complex".
He added that he "does not exclude" interviewing Sepp Blatter, president of global football's governing body FIFA, and Jerome Valcke, FIFA's secretary-general.
The FBI and Swiss authorities are investigating the bidding for the 2018 and 2022 World Cups that were awarded to Russia and Qatar respectively.
Lauber is prepared for either country to be stripped of the hosting rights if new evidence proves wrongdoing.
"I don't mind if this has some collateral [damage] somewhere else," he said before declining to discuss a timetable for the case, which targets "criminal mismanagement and money laundering" in the bidding process.
"This is a dynamic process. It could really go everywhere and that is why I don't want to tell you which direction I put my focus."
Blatter is also a target of a separate investigation of bribery and racketeering led by US federal agencies, who are working with Swiss authorities.
The American case alleged that senior FIFA voters received $10m in bribes to support South Africa's successful bid for the 2010 World Cup.
Valcke has been linked to transferring the money from a FIFA account on behalf of South African officials, Al Jazeera reported.
FIFA was thrown into crisis on May 27 with dual raids in Zurich on a luxury downtown hotel and its own headquarters two days before its presidential election.
Two FIFA vice presidents, Jeffrey Webb of the Cayman Islands and Eugenio Figueredo of Uruguay, were among seven football officials arrested by Swiss police acting on a US request. They face extradition to the US.
On the same day, data and documents were seized at FIFA's offices for the Swiss case. More evidence was collected from FIFA last week.