The report claims that the Black Eagles are keen on bringing in the Ivory Coast international to play alongside Germany international Mario Gomez going into the Champions League next season.
Bony joined City on a £25 million move from Swansea City last season but failed to settle in despite scoring nine goals and providing six assists last season.
It remains to be seen if Pep Guardiola has plans for Bony next season but Swansea are also reported to be keen on signing the powerful striker.
Besiktas would be able to offer Bony first-team action and Champions League football next season in one of the most exciting sides in the Turkish league.
The Black Eagles will be playing in their new Vodafone Arena stadium next season and are expected to make a marquee signing to coincide with their new ground.