But his demand has put him on a collision course with Chelsea, who would prefer to delay the procedure.
The recovery period is up to five weeks and boss Avram Grant is considering asking Drogba to play on until Christmas before going under the knife.
It is possible he could play on with the aid of painkilling injections until the transfer window opens in January and a replacement - most likely Nicolas Anelka - could be bought.
But Drogba, who is captain of Ivory Coast, is determined to play in the African tournament which starts on January 20.
A close friend of Drogba told The Sun: "Didi understands the problem this potentially causes for the club but they have known for a long time about the African Nations Cup.
"He has a responsibility to his country and is captain of the team. Didi is very passionate about playing for the Ivory Coast and does not want to let them down."