Bony, a 26-year-old from Ivory Coast, joined Swansea from Dutch club Vitesse Arnhem in 2013 for a Swansea-record deal worth £12 million. In his final season with the Dutch side, Bony scored 31 goals in 30 league appearances, earning him Dutch Player of the Year honors.
During the 2014 calendar year, Bony was the Premier League's top scorer with 20 goals. In 22 appearances this season, he has nine goals, including an injury time strike against Queens Park Rangers on New Year's Day that tied the match and helped Swansea earn a draw.
In 70 games with Swansea, Bony has scored 34 goals.
The deal is reportedly £25 million up front with £3 million in performance-related bonuses.
According to the BBC, Manchester City expects no problems with personal and medical terms, despite Bony's hiatus from Swansea over the next three weeks to take part in the Africa Cup of Nations.
The defending Premier League champions are currently in second place with 47 points, two behind Chelsea.