Melo has been a long-term target for the Arsenal manager Arsene Wenger who sees the Brazilian international as the man to bolster his defensive options in midfield.
Inter Milan have also been linked with Melo in the past week. La Gazzetta dello Sport have claimed Inter are close to agreeing a €5 million deal for the former Juventus star.
The 31-year-old joined the Istanbul giants on a permanent deal from Juventus worth £3 million last summer after playing the previous two seasons on loan at the Lions.
Galatasaray ended the 2014-15 season with 77 points following their 1-1 draw with Rizespor in their final game of the season.
The Istanbul giants will qualify directly for the Champions League group stages next season.