The Germany international returned to his homeland to consult a specialist in Munich after suffering a recurrence of inflammation around the metatarsal bones in his right foot, with the decision subsequently taken to operate.
However, Lazio are confident of having the ex-Bayern Munich forward available after the international break, with the away fixture against Atalanta on October 20 pencilled in as a potential date for his return.
Lazio have made a solid start to their Serie A campaign, having won three and lost two of their opening five fixtures, and sit sixth in the league.
Klose, 35, has found the target once in his four games and Lazio will be keen to have him available soon to boost their striking options.
The Rome outfit visit Sassuolo on Sunday, who will be buoyed by gaining their first point of the season in a 1-1 away draw with high-flying Napoli on Wednesday having been thrashed 7-0 by Inter the weekend before.