The former Chelsea striker had been in talks with officials from Gala over the past 24 hours over a move to Istanbul.
Following those discussions and a medical test, the 34-year-old has opted to leave Chinese side Shanghai Shenhua.
"Big clubs can reach for higher targets with bigger goals," said the club's president Unal Aysal. "Drogba said that he was very happy to play for a big club like Galatasaray in the future. Drogba met our coach and decided to come to Galatasaray."
Drogba is currently part of the Ivory Coast squad that is competing at the Africa Cup of Nations.