The Ivory Coast international has struggled in Manchester since moving from the Liberty Stadium for £30m in 2015.
But according to The Sun, Bony will be given a chance to relaunch his career at Swansea under Paul Clement, who is willing to spend £13m on the striker.
The Welsh club will spend some of the £45m fee they will receive from Everton for Gylfi Sigurdsson, who is having a medical today at the Toffee’s Finch Farm training ground.
The 28-year-old was prolific for Swansea during his 18 months in South Wales, scoring 35 goals in 70 games.
However his move to Man City was littered with substitute appearances and he only scored 10 goals in his 46 games, before being sent on an unsuccessful loan spell at Stoke last term.
He scored just two goals for the Potters, both against Swansea, and slated Mark Hughes for his treatment during his time at the Bet365 Stadium.
Bony told BBC World Service: “I want to know why I’m not playing. It is really crazy when the people say you are fine and that everything is okay, but you don’t play. I can say it is like maybe to p*** you off.
“I have asked him and he says I am training well, my attitude is good and I don’t need to change that but why I don’t play, I don’t know.
“I am not playing for no reason and since he spoke about the Chinese clause in my contract I don’t play - it’s really strange.”
If Bony, who is on £130,000-a-week at Man City, agrees personal terms with the Swans, he will become the fifth summer signing for Swansea.
He will compete with on-loan Chelsea striker Tammy Abraham and Fernando Llorente for a spot in the side.
