The Costa Rican international is out-of-favour at the Emirates Stadium and has Theo Walcott, Olivier Girous, Alexis Sanchez and Danny Welbeck ahead of him in the pecking order.
Campbell, who joined the Gunners in 2011 from Deportivo Saprissa, was only handed a work-permit to play in England last summer. He spent the previous three campaigns on loan in Spain with Celta Vigo and in Greece with Olympiakos before returning to north London last summer.
The 23-year-old had an impressive loan spell with the Greek champions, before going on to star in the 2014 World Cup for Costa Rica. But following his return to Arsenal, he found first-team chances hard to come by, making just four appearances during the first half of the 2014/15 campaign.
He spent most of his time as a substitute bench, and spending a total of just 39 minutes on the pitch during his time at the club. The lack of football with the Gunners saw him join Villarreal on loan during the January transfer window.
However, the Costa Rican forward failed to impress in the La Liga managing just one goal in 22 appearances for the Yellow Submarines and it is believed that Arsene Wenger is ready to sanction his move to the Turkish club, with the 12-time FA Cup champions receiving a fee of around €5m (£3.5m, $5.5m) to part with the player.
The Turkish publication claims that Besiktas have already reached an agreement with the player with regards to personal terms, with the club's manager looking for a replacement for last season's top scorer Demba Ba, who has joined Chinese club Shanghai Shenhua.