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According to a report from Turkish outlet Fanatik, the Istanbul giants have agreed personal terms with the Costa Rica international.
Campbell is currently on international duty with Costa Rica and will feature at this summers 2018 World Cup in Russia.
The 25-year-old scored during his sides 3-0 win over Northen Ireland earlier this week.
Campbell will be no stranger to Galatasaray fans after being linked with a move to Istanbul for the past three years.
The Costa Rica international is expected to leave Arsenal this summer after spending his last six season out on loan at Betis, Sporting Lisbon, Villarreal, Olympiacos and Lorient.
The former Saprissa star has made just 23 appearances for Arsenal in over six years at the club.
Galatasaray lifted the Super Lig title for the 21st time this season and will compete in the Champions League.
The Lions have made strengthening their attack a priority ahead of their European campaign.