Premier League champions City have moved for Bony, who has scored 34 goals in 70 appearances since a £12m move from Holland’s Vitesse Arnhem in summer 2013, as all of their senior strikers are sidelined by injury.
Top scorer Sergio Aguero is nursing a knee ligament problem, Stevan Jovetic has a hamstring injury and Edin Dzeko is also hurt, leaving City to play midfielders as makeshift forwards.
Bony, 26, is due to jet out to his country’s training camp for the imminent Africa Cup of Nations today but could complete a move before then if all remaining formalities can be thrashed out.
But if Ivory Coast go all the way to the final of the tournament on 8 February he looks certain to miss City fixtures against Everton, Arsenal, Middlesbrough, Chelsea, Hull and Stoke.
Bony signed a new contract in November which removed a clause allowing him to leave for £20m but is believed to have told Swansea he is keen to leave if wanted by a Champions League side.
City were limited to an outlay of £47m during the summer transfer window as part of their punishment for breaching European financial fair play rules, but that cap has now expired.
Manuel Pellegrini’s team are currently level on points with Premier League leaders Chelsea.