Bony has struggled to make an impact at City since arriving from the Swans in a $36.7 million deal two years ago, managing just eight goals in 46 appearances.
The Ivory Coast ace does not figure in the plans of boss Pep Guardiola moving forward and Swansea are keen to take him back to the Liberty Stadium for a fraction of the price they sold him for.
City are happy to let him move on as they look to recoup some cash before he falls out of contract in 12 months, and they will be pleased to hear that Besiktas have also entered the running.
However it is bad news for the Swans as reports claim that Besiktas will be able to get closer to the $157,500 a week the player currently picks up at the Etihad Stadium.
The 28-year-old will have a choice to make and he could be tempted by the cash and a fresh challenge in Turkey over a return to south Wales.