The deal is believed to be £25m cash upfront with a further £3m in performance-related add-ons, while City expect to finalise the 26-year-old’s move to the Etihad Stadium within the next 24 hours.
Bony is currently on Africa Cup of Nations duty ahead of the tournament in Equatorial Guinea, which gets underway on 17 January, and will not be able his City debut for at least three weeks.
The former Vitesse Arnhem forward has scored 37 goals across all competitions since joining Swansea for a club-record fee of £12m in 2013, with his last strike an injury-time leveller against QPR on New Year’s Eve.
Bony’s tally of 20 made him the Premier League’s highest scorer in 2014, while his nine goals in 22 appearances have triggered the attentions of City boss Manuel Pellegrini who has been keen to add to his stable of striking talent.
Pellegrini was forced to play James Milner as an emergency forward over the festive period after injuries to his three mainline hitmen, although 14-goal Sergio Aguero made his return from injury on Saturday.
City, meanwhile, are currently under investigation by European governing body Uefa over the logistics of their parent company, City Football Group, and their subsidiaries sharing the club’s sizeable losses.
The club were fined £16m and set a £49m net transfer spend restriction this term having failed to comply with Uefa’s financial fair play rules last season.
After City announced a £23m loss for 2013-14, a reduction of around half on the previous year, Uefa is examining club structure and whether any subsidiary companies are absorbing losses while executing City business.