Chelsea owner Roman Abramovich was ready to offer Drogba a deal until the end of the season, with an option to extend further. Initially it was clear that Drogba is set for an emotional return to the Bridge as he joined his former teammates to practice for the African Cup of Nations.
Drogba, 34, who left Chelsea after helping them lift their first ever Champions League trophy, to join Chinese club Shanghai Shenhua for a two-and-half-year deal worth nearly £300,000-a-week.
The Chelsea legend, currently on national duty at the African Cup of Nations, was heavily linked with Juventus during the January transfer window.
However, Galatasaray confirmed on their website that talks with the players representatives over a transfer have begun.
Drogba left Chelsea just six months ago, putting an end to a glittering eight-year career at Stamford Bridge, but his spell in China created rumours that the player is likely to make a quick return to Europe.
Drogba, who won three Premier League titles, four FA Cups and ended his Chelsea career with the Champions League title, was offered a year's extension to stay at the Bridge, but he decided to end his career after winning Europe's elite competition.
With the return to Europe, he will guarantee Champions League football by signing for Galatasaray, who face Schalke in Europe next month.
The cash-rich Turkish side signed former Champions League winner Wesley Sneijder from Inter Milan last week, and are expected to complete the signing of Drogba this week.
Drogba has made 341 appearances for Chelsea scoring 157 goals which includes 100 goals in the Premier League.
Former Chelsea star and Drogba's team-mate at Shenhua Nicolas Anelka has joined Italy Serie A champions Juventus on a emergency loan until the end of the season.