Turkish broadcaster CNN Turk reported late on Sunday the Ivory Coast captain and former Chelsea player had already signed a one-and-a-half-year deal with the Turkish club but those reports could not be confirmed.
The 34-year-old, who is on Africa Cup of Nations duty with his country, moved to China after helping Chelsea win the Champions League last May.
Drogba applied for permission to leave on loan before the January transfer window but world football's ruling body FIFA refused the request.
The striker returned to Chelsea to train at his former club in a bid to stay sharp for the Nations Cup, fuelling media speculation of a return to Europe.
If Drogba does move to Turkey, the transfer will come less than a week after Netherlands midfielder Wesley Sneijder joined Galatasaray from Inter Milan.
Galatasaray have been drawn to meet German club Schalke 04 in the last 16 of the Champions League next month.
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