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 Black Eagles are looking to bolster their attacking options after offloading last seasons top goalscorer Demba Ba to the Chinese club Shanghai Shenhua.
Campbell has been transfer listed by Arsenal after spending the last few seasons out on loan according to the Daily Mail.
The Costa Rican international was expected to feature for Arsenal last season after Gunners boss Arsene Wenger insisted that the Campbell remained at the club.
The 23-year-old was however, not given a chance and subsequently loaned out to Real Sociedad over the winter transfer window despite interest from Galatasaray.
Beşiktaş finished the Turkish Super Lig season in third place and will play in the Europa League next season.