Bony moved to City from the Swans for £25m in January 2015, but struggled to get a consistent run in the first team under former Blues boss Manuel Pellegrini and has never featured in Pep Guardiola's plans.
The Ivorian spent last season on loan at Stoke City, where he scored two goals in 11 games, and has had a number of offers during the current transfer window, including a late bid from Verona.
But City have now accepted a bid from the Welsh side and the striker will complete the move later on Thursday, having earlier trained in Manchester.
Swansea have long been interested in the 28-year-old but a move for him was depending on Fernando Llorente being sold.
Chelsea had been interested in Llorente but it is now Spurs who are in pole position to secure his signature in a £15m deal.
Former Vitesse striker Bony scored 35 goals for Swansea in his 18 months at the club, but only managed 11 for City since joining in January 2015.