The Swans are considering re-signing the Ivorian, who left the Liberty Stadium to join Manchester City in 2015 for a fee worth up to £28m.
Bony, 28, managed only eight goals in 46 appearances for City - more than half of which came as a substitute - and spent last season on loan at Stoke.
His contract at the Etihad Stadium expires in 2019.
Bony is believed to earn more than £120,000 a week at City but, having qualified for the Champions League, Besiktas have money to spend.
They won the Turkish title last season and their signings this summer include Portugal defender Pepe, who joined after his contract at Real Madrid expired.
Swansea's potential move for Bony, meanwhile, could depend on the future of their top scorer last season, Fernando Llorente.
The 32-year-old, who is currently sidelined with an arm injury, has been a transfer target for Chelsea, having worked with the Premier League champions' manager, Antonio Conte, at Juventus.
Swansea have yet to receive a bid this summer for the Spain international, who scored 15 goals last season to help the Welsh club avoid relegation from the top flight.
The Swans did reject offers for Llorente during the January transfer window, and a move for Bony could be seen as insurance should Llorente leave.
Swansea City face competition from Turkish club Besiktas for the signing of striker Wilfried Bony.