The Ivorian joined the Istanbul giant in January on an 18-month deal, but the club is confident he will opt to extend his stay in Turkey.
The former Chelsea and Marseille forward joined the Turkish side in January on a year-and-a-half deal which is due to run out in June 2014.
Aysal is ready to offer fresh terms to Drogba who has scored four goals in 10 games for the Super Lig champion this season, and is confident the 35-year-old will opt to stay in Istanbul.
"Drogba has talked about issues related to the future," Aysal told GSTV. "He wants to extend his career as a player. I told him that, at Galatasaray, you can play as long as quality is evident.
"We are definitely going to offer him a [new] contract. I also do not think he will reject it."
Galatasaray currently sits second in Group B of the Champions League on four points, with a home tie against Juventus and visits to Real Madrid and Copenhagen to come.
In the Super Lig, Roberto Mancini's men sit seventh after eight games - nine points behind league leader and arch-rival Fenerbahce.