The Ivory Coast international striker is set to miss Saturday's Premier League clash with Sunderland and will probably not be required for next week's Champions League clash with Valencia as the Blues are already assured of qualification for the next phase.
Drogba's own priorities are understood to be slightly different however as he is keen to be fit in time to lead the Ivory Coast into the African Nations Cup competition, which starts next month.
An operation performed now would give him time to recover for that but Grant insisted that not only he did not know whether one would be required but also that he believed the player was still keen to represent his club during a key period of the season.
He said: "We are keeping an eye on the situation but we don't know yet. We will know better in the next few days.
"We are doing everything with Didier. In the last one or two months he has had a problem but he always pushed to play and is pushing to play even now.
"But we need to take care of the health of the player and we will see what is good for him. At the moment we can only keep an eye on the situation and then we will know what will happen.
"It depends on how he feels. If he feels good he will play."