The Lazio striker went under the knife after suffering a relapse of an inflammation around the metatarsal bones in his right foot.
He is set to return after the international break, with the target Lazio's game against Atalanta on October 20.
Klose told Bild: "The surgery has been OK. It had to be done. I've played many weeks with pain.
"I will do everything to come back as soon as possible for Lazio and for my national team too."