Galatasaray said on its website Monday that Drogba had agreed to an 18-month deal. Drogba has spent this season in China with Shanghai Shenhua but is currently with Ivory Coast at the Africa Cup of Nations in South Africa.
The 34-year-old left Chelsea for China last year after helping the west London side to its first Champions League title.
Galatasaray, the Turkish champion, already signed Inter Milan star Sneijder this month as it looks to boost its squad after reaching the last 16 of the Champions League, where it will face Schalke, and it has also been heavily linked with Manchester City defender Kolo Toure.
"The opportunity to play for this great club was an offer that I could not turn down, Drogba said through his representatives, Sports PR Company, on Twitter. "I am looking forward to playing in the Champions League again, against the best clubs in Europe."
Drogba's move to Turkey brings to an end a six-month stint in China.
Despite his protests to the contrary, reports persisted that Drogba's move to China had not worked out, largely because of boardroom wrangling.
FIFA rejected a special request from the striker for him to join a club on loan outside the transfer window, amid speculation linking him with Juventus, AC Milan and Liverpool, among others.
Last month, Drogba returned to train with his old club during the Chinese offseason in preparation for the Africa Cup of Nations.
Drogba had signed a two-and-a-half-year deal at Shenhua, netting him around $395,000 a week and reuniting him with Nicolas Anelka.
Anelka is on the verge of joining Juventus on a deal until the end of the season.
Information from The Associated Press and Press Association was used in this report.