The former-Stamford Bridge striker is reportedly under consideration after Galatasaray sacked Terim following his refusal to sign a contract extension.
Also on the shortlist is former-Chelsea boss Roberto Di Matteo and ex-Manchester City manager Roberto Mancini.
Drogba's standing in the dressing room has seen him elevated to candidate for the job despite his lack of managerial experience.
Reports in Turkey suggest Galatasaray could look to the Ivorian as a caretaker boss as they continue their search for Terim's replacement.
An article in Portuguese newspaper A Bola claims the ex-Chelsea forward has already been approached by the Galatasaray hierarchy as they sound out his interest in the role.
The 35-year-old joined Galatasaray after a brief stint in China, and although he's never managed before he could be tempted by a player/manager position.
The Turkish giants currently sit 11th in the league, and were humiliated 6-1 at home to Real Madrid in their first Champions League group stage match of the season earlier this month.