Campbell who joined Arsenal in 2011 from Deportivo Saprissa has spent the past three seasons out on loan at numerous clubs. The Costa Rica international is expected to leave this summer after falling out of favour with the Gunners boss Arsene Wenger.
The report goes onto state that Besiktas have agreed personal terms with the young striker and have struck a €5 million deal with Arsenal for the transfer fee.
Besiktas manager Senol Gunes has stated that he wants to see Campbell at the Istanbul based club next season.
The Black Eagles are looking to bolster their attacking options after offloading last seasons top goalscorer Demba Ba to the Chinese club Shanghai Shenhua
Campbell struggled for playing time last season, the 23-year-old forward only played 40 minutes of Premier League football for the Gunners.