The 30-year-old is expected to be ruled out for at least two weeks, and thus will miss Germay's next two World Cup qualifiers against Liechtenstein and Wales.
Klose was hurt in Bayern's 3-0 win at VFL Bochum in the Bundesliga on Saturday after stepping into a small hole on the pitch.
Major Setback
Bayern boss Jurgen Klinsmann believes his German countryman will be sorely missed, given that duo Luca Toni and Franck Ribery are yet to return from injury.
"This is a bitter diagnosis and, of course, a major setback," Klinsmann, who now has only one available striker in Lukas Podolski, told Bild.
However, the German champions could be buoyed by the return of Italy hitman Toni, who seems set to be named for Bayern's upcoming game against Karlsruhe.
"This is a very difficult situation now," Klinsmann added. "I have a good feeling that Toni may be fit for the next match."