"After taking legal advice we decided to take this step," said Schalke sports director Horst Heldt, adding they had the right to do so within 24 hours from the end of Wednesday's first-leg encounter.
"It is our duty to demand a check of eligibility for Drogba and to reserve our rights in this matter," he said.
Ivorian striker Drogba joined the Turkish club in late January for 18 months from Shanghai Shenhua. The Chinese club had initially tried to block the transfer saying he had violated the terms of his contract.
Drogba had moved to Shanghai Shenhua last June after winning the Champions League 2012 with Chelsea but cut short his stay in China amid reports of unpaid wages.
His deal with Galatasaray includes transfer fees totalling 6 million euros ($8 million) and payments of 15,000 euros per match, plus signing fees of 4 million euros.
"There is the suspicion that he received his eligibility after the end of the deadline to register the player for the UEFA Champions League," said Schalke finance boss Peter Peters.
Schalke said they had informed the Turkish club of their actions.
"It is now up to UEFA to decide and Schalke will respect this process," said Peters.
(For more sports news and updates, follow Ahram Online Sports on Twitter at @AO Sports and on Facebook at AhramOnlineSports)