Turkey hitman Tosun is out of contract at Everton at the end of the season and is expected to move on from Goodison Park.
His agents are looking at potential options and Tosun is an attractive prospect due to his impending free agent status.
Besiktas have long shown interest in signing Tosun, a player who has had two spells with the club, and now they have put him at the top of their transfer wish list, according to Turkish outlet Fotomac.
The Black Eagles have intensified contacts with Tosun as they look to tempt him back to Turkey.
Besiktas have Michy Batshuyai, on loan from Chelsea, leading the attack, but he looks unlikely to stay at the club beyond the end of the campaign.
The Belgian striker has scored eleven times in 31 appearances for Besiktas so far this season.
Besiktas sit in sixth spot in the Turkish Super Lig standings, just two points from third place with 27 games played.
