The Dortmund youngster will continue his career in Istanbul, having signed a contract worth €700,000 a year until the summer of 2018
The Germany youth international, who is of Turkish descent, has spent his entire career to date at Signal Iduna Park, but made only one Bundesliga appearance.
Gunter has sought to continue his career in Istanbul, though, BVB have the opportunity to buy him back for a fee of €7m for the 2015-16 season.
"The professional footballer Koray Gunter and Borussia Dortmund have both reached an agreement over the player's transfer," a Galatasaray statement read.
Galatasaray will pay BVB an additional €150,000 for every 20 professional games Gunter plays, and the player will pick up an annual salary of €700,000.
The 19-year-old represented Turkey at Under-16 level, but switched his allegiance to Germany for the Under-17s and Under-19s.