The International Football Association's investigation by Fifa showed that the project directly involved the former chairman Sepp Blatter and people around the construction of the Haus zur Enge museum, which was completed in 2016.
At the time, the museum was said to have cost the equivalent of SEK 1.2 billion of Fifa's money to build.
But the investigation shows that the total construction cost landed at the equivalent of SEK 4.7 billion, of which SEK 1.3 billion was used to renovate and furnish a building that Fifa does not own.
"Had no other choice"
According to the agreement that was written, Fifa is also stuck in an unfavorable lease agreement that will cost Fifa SEK 3.4 billion until it expires in 2045.
"The investigation revealed a number of suspicious circumstances and mistakes, some of which may be criminal and which therefore need to be properly investigated by the appropriate authorities," Fifa's acting secretary general Alasdair Bell told the association's website.
"Therefore, we came to the conclusion that we had no choice but to go to court, not least because the current management of Fifa is responsible for the organization we want to live up to, even if those before us unfortunately failed to do so."
Blatter replies: "The accusations are unfounded"
Fifa's ethics committee will also look into the matter.
Sepp Blatter was fired in 2015 after being involved in a major corruption scandal.
Blatter denies any wrongdoing.
"The allegations are baseless and we categorically reject them," his lawyer Lorenz Erni was quoted as saying by Reuters.