Eurosport TR reported that Beşiktaş want a €7 million buy-out option included in the deal.
The Black Eagles are in the market for a striker after selling Demba Ba to Chinese giants Shanghai Shenhua for €14 million a fortnight ago.
Crystal Palace and Swansea are also reported to be interested in signing the 28-year-old striker.
Chelsea boss Jose Mourinho is believed to be reluctant on letting the France international go but could be persuaded now the Blues have Falcao.
The Istanbul giants have also been closely linked with a move for Arsenal forward Joel Campbell.
Blues boss Jose Mourinho, however, blocked Remy’s exit as he wants to keep experienced back-up at the club.
The 28-year-old provided two assists and scored nine goals in 29 appearances for the Blues in all competitions.