The fire broke out about 2 a.m. but was not reported until hours later, Minnesota Public Radio reported.
Winona Fire Chief Curt Bittle said by the time crews arrived, the Islamic Center of Winona's roof had already collapsed, the report said.
Winona Police Chief Paul Bostrack said investigators had not determined where the fire originated and had not seen evidence of arson.
"At this point we have nothing that indicates that," he said.
Ahmed El-Afandi, a co-founder of the 20-year-old center, said the center had not received any threats.
"Nobody [in the community] has given us any trouble," said El-Afandi, a retired professor of political science at Winona State University.
The Bureau of Alcohol, Tobacco, Firearms and Explosives sent investigators to the scene, but ATF spokesman Robert Schmidt said there is "no indication at this time" the fire was intentionally set, the Minneapolis Star-Tribune reported.
At least 16 people were burned out of apartments.