The 35-year-old suffered the injury in Lazio's 1-1 draw at Parma on Sunday and looks set to be sidelined until December.
Klose will miss Germany's friendlies against Italy and England on Friday and Tuesday respectively despite travelling with the squad.
"His (Klose's) shoulder is painful, as he had already suffered dislocations in the past," Lazio doctor Roberto Bianchini said on Tuesday.
"He has left for international duty and we are in contact with the German medical staff, so we await news. In any case, he will have to rest for around 20 days."
There was good news for Lazio with defender Abdoulay Konko (thigh) available and Ederson and Senad Lulic close to returns.
"Abdoulay Konko is available, while Ederson has just a tiny muscular twinge and Senad Lulic is recovering according to plan," Bianchini said.
"Giuseppe Biava will have an x-ray today and visit a specialist tomorrow. He put on football boots and seemed to react positively, so we hope he will try to be back within a couple of weeks."
