The Lions will enter talks in January with the Costa Rican international who has struggled for playing time this season.
The Istanbul giants were closely linked with Galatasaray over the summer but Arsenal boss Arsene Wenger insisted that the 23-year-old forward stayed.
Campbell has played just 398 minutes of football this season over the span of seven appearances.
Galatasaray currently lack options going forward and are looking to boost their attacking options over the winter transfer window.
Campbell has just a year remaining on his contract meaning Galatasaray could enter direct talks with the 23-year-old should Arsenal decide not to extend his current deal.
Galatasaray are the reigning Turkish Super League champions and are currently competing in the Champions League group stages which would mean Campbell would be cup tied should the Lions progress to the next round.