The 35-year-old was replaced in the 84th minute of the game at the Stadio Ennio Tardini on Sunday and now faces further tests to determine if he will be fit to play as Germany starts its preparations for next year's World Cup.
Klose's third season with Lazio has been heavily affected by injury, with the former Bayern Munich man forced to undergo surgery in September because of a metatarsal injury.
Germany coach Joachim Low could also be without Arsenal pairing Mesut Ozil and Per Mertesacker due to a virus that kept the latter out of the Premier League side's 1-0 defeat to Manchester United.
The duo will not travel to Munich to link up with the squad, and it remains to be seen if they will be able to recover in time to face two of their prospective World Cup rivals.
Vice-captain Bastian Schweinsteiger has already been ruled out as the Bayern midfielder faces another procedure on a persistent ankle problem.
Germany takes on Italy at San Siro on Friday in what will be Low's 100th game in charge, before meeting England at Wembley next Tuesday.