The Bundesliga outfit may contact UEFA to question whether Galatasaray's Cote d'Ivoire international was eligible for their Champions League clash.
The Gelsenkirchen club believes the Cote d’Ivoire international was not properly registered to play in the European competition for Fatih Terim’s side.
FIFA ratified the switch from Shanghai Shenhua despite the Chinese club's protests, but it is possible Schalke is protesting over a separate license-to-play issue.
"There are doubts about the validity of the permission to play for Drogba in the Champions League," the Bundesliga club announced on its official Twitter account. "Schalke reserves its rights and is looking into this."
The former Chelsea striker joined Galatasaray on an 18-month contract in the last few days of the January transfer window, leaving Shanghai Shenhua on a free transfer after the club had been deemed not to have paid his wages.
Schalke general manager Horst Heldt claimed on Thursday "it would be irresponsible not to [launch an appeal]."
Drogba did not have much of an effect on the scoreline on Wednesday, as Burak Yilmaz's opening goal was cancelled out by Jermaine Jones' equalizer, and the two teams fought the remainder of the match on level terms, with Schalke taking the advantage into the return leg via Jones' away goal.