Drogba will sign his deal when the Ivory Coast's run at the Africa Cup of Nations comes to an end - their next game is a quarter-final in Rustenberg on Sunday.
The 34-year-old left Stamford Bridge last summer for China in the wake of leading the Premier League club to UEFA Champions League glory.
Drogba's signing is the second major deal Galatasaray have completed in the January transfer window, having brought Dutch midfielder Wesley Sneijder to the club last week.
According to reports, Drogba will earn 10 million euros over the 18-months of his contract with a 15,000 euro bonus for each match he plays.
Drogba said, through his representatives Sports PR Company, on Twitter: "The opportunity to play for this great club was an offer that I could not turn down.
"I am looking forward to playing in the Champions League again, against the best clubs in Europe."