The Colombian international is currently on-loan at the Blues but Besiktas had been in talks and the transfer committee were keen to make him the first transfer of the season.
Falcao joined Chelsea on loan but struggled to settle in scoring just one goal in 12 appearances for the reigning champions this season.
Gunes has however, told the board that he does not want to pursue Falcao who has seen a drop in form over recent years.
Mario Gomez has has been in fantastic form for the Black Eagles scoring 27 times for the champions but he is on-loan from Fiorentina and Besiktas are yet to sign him on a permanent deal.
Gomez has 25 league goals and has attracted the interest of several Premier League and Bundesliga sides.
Besiktas are currently in talks trying to sign Gomez for at least another season.