The Turkish newspaper state that the North London based side have green lit a €3.5 million bid for the 22-year-old.
Besiktas manager Senol Gunes has stated that he wants to see the Arsenal forward Joel Campbell at the Istanbul based club next season.
According to a report in the Vatan newspaper, Gunes has instructed the club to sign the 23-year-old after inspecting the scout reports on the player.
The Istanbul based side are looking to improve their attacking line-up after selling last seasons top goalscorer Demba Ba to Shanghai Shenhua.
Campbell has been transfer listed by Arsenal after spending the last few seasons out on loan according to the Daily Mail.
Beşiktaş finished the Turkish Super Lig season in third place and will play in the Europa League next season.