Drogba went under the knife on Saturday to correct an ongoing knee injury with the club claiming the operation went without mishap.
Grant, speaking after the 2-0 victory over Sunderland, refused to give a date for his comeback but suggested he may be on club duty in January - and not at the African Nations Cup.
He said: "The future is not easy without Drogba but I didn't come to this club for an easy life. We didn't want him to be injured.
"The operation looks like it went well. Drogba is in good shape and we want to keep him like that.
"He had a problem with his knee, we waited a few days and then he knocked it again last night.
"He needed the operation and there was nothing we could do about it.
"We need to wait and see how long he'll be out for. We'll know exactly over the next two or three days."
When asked about Drogba's impending call-up by the Ivory Coast in January, Grant replied: "Are we losing him to the African Nations Cup? We want him to be well.
"If his knee is OK and he plays in the competition then good, but if he can also play for Chelsea then that is also good.
"What's important now is that he feels good."