The 31-year-old was unlikely to remain at the Liberty Stadium this summer following his return from a season-long stint with Marseille, where he netted 20 goals in 31 league appearances.
Gomis could only manage 17 goals in 71 games for the Swans and was not part of boss Paul Clement's plans for the new season, opening the door for Galatasaray to push through a deal.
Turkish giants Galatasaray confirmed on their Twitter account that official talks are now under way, while Gomis took to Instagram to post a farewell message to City supporters ahead of completing his switch.
Bafetimbi Gomis, who was announced to have started negotiations in order to complete his move to Galatasaray, has today landed in Istanbul! pic.twitter.com/Pa9raOnGHy— Galatasaray EN (@Galatasaray) June 28, 2017
"I had come with a very specific goal, to help the club qualify for Europe and score 20 goals, and this happened," he said. "Today a page turns for me.
"I have made the choice to leave for a new experience, I leave with a heavy heart because I leave friends, brothers, a stadium, a city, a family, great men."
Swansea are expected to receive a fee of around £2.5m for the striker, who arrived as a free agent from Lyon three years ago.