However, City will include a £17 million release clause in the Ivory Coast international’s loan deal in a bid to recoup some of the £28 million spent on him last year.
Bony’s future is unclear following the appointment of Pep Guardiola and he has yet to make a decision regarding the 27-year-old’s place in the squad.
The report states that Galatasaray are keen on the powerful striker and the clubs director of football Cenk Ergun has flown to Manchester in order to begin negotiations.
The Turkish giants will pay £3m up front for Bony and have the option to wrap up a permanent deal next summer.
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.