Campbell is expected to leave the Gunners after failing to settle in and the Lions have been closely linked with the Costa Rican international over the past few seasons.
The 23-year-old has been transfer listed by Arsenal after spending the last few seasons out on loan according to the Daily Mail.
The Costa Rica international received a number of offers last summer but Gunners boss Arsene Wenger wanted Campbell to stay at the North London based club before eventually letting him go on a temporary move.
Campbell was loaned out to Villarreal at a later stage in the season after failing to establish himself in the Premier League.
Galatasaray are currently second in the Turkish Super League and have signed a number of former Arsenal players over the last few years including Lukas Podolski and Emmanuel Eboue.