The Lions are reported to have added Campbell to their shortlist as they continue efforts to add depth to their forward line.
The Istanbul giants are currently suffering a striker shortage due to Burak Yılmaz being out injured. Galatasaray currently only have Lukas Podolski, Umut Bulut and youngster Sinan Gümüş available.
Arsenal could be tempted to cash in on Campbell as he has just a year remaining on his current contract which runs out in the summer.
Campbell has struggled for playing time this season making just three appearances in all competitions but playing just a total of 95 minutes and not providing a single assist or goal.
This is not the first time Galatasaray have made the Gunners an offer for Campbell the Lions were closely linked with the Costa Rican international over the summer and previous transfer windows.