The Turkish Super Lig broadcaster report that the Black Eagles had a loan bid turned down as Chelsea want to sell Remy on a permanent deal.
The Istanbul giants have made an enquiry regarding the transfer fee Chelsea would be looking to accept for the French striker.
Remy could fall further down the pecking order at Chelsea if they manage to sign Everton striker Romelu Lukaku.
Besiktas meanwhile would be able to offer Remy Champions League and first team football next season if he were to join.
The Eagles are in the hunt for a new striker after Mario Gomez revealed he will not be returning to the club.
Gomez joined Besiktas on loan last season from Fiorentina and ended the season as their top scorer. The experienced striker played a key role in the Eagles winning the league title and managed to earn a place back in the Germany national team as a result.
The 31-year-old scored twice at Euro 2016 this summer playing an important role in Germany reaching the semi-finals.