The Lions are reported to have made City an enquiry regarding the availability of Bony next season.
The future of Bony is unclear following the appointment of Pep Guardiola and he has yet to make a decision.
Bony joined City on a £28 million move last season but he struggled to settle in and has been linked with a City exit.
Galatasaray are currently under supervision from FFP and are serving a one-year UEFA competition ban.
The Lions won the Turkish Cup last season but had a disappointing league campaign and plan to restructure the squad as a result.
Turkish Super Lig champions Besiktas also recently made an enquiry regarding whether the Ivory Coast international will be available on loan next season.
The Black Eagles will be able to offer Bony Champions League and first team football if he were to consider making the move.