The midfielder, who never settled at Real Madrid under coach Jose Mourinho, has already met up with his new team-mates for their close-season preparations
The 29-year-old has been tipped to leave the Blancos since the end of last season after failing to make his mark at the Santiago Bernabeu following his free transfer from Bayern Munich.
Neither side have thus far announced the transfer, but Galatasaray coach Fatih Terim told Galatasaray TV on Thursday that the player has joined up with his side for pre-season.
"Hamit is an important transfer," Terim told the channel. "I would like to welcome him personally.
"We have got a player who will be useful for many years. All of our club will benefit."
The Turkey international, who will start training with the rest of the squad on Friday, also commented on the move, saying: "I believe we have done everything properly. There are no more issues.
"Tomorrow, I will start working to achieve our targets."
Hamit began his professional career with Wattenscheid and also played for Schalke before joining Bayern in 2007.