Swansea manager Garry Monk previously said it would take an "astronomical" fee to prise the Ivory Coast striker, 26, away from the Liberty Stadium.
He was the Premier League's top scorer in the calendar year of 2014 with 20 goals.
Bony joined Swansea for a club-record £12m from Vitesse Arnhem in 2013.
The fee breaks the Welsh club record sale of £15m that Liverpool paid Swansea for midfielder Joe Allen in August 2012.
Bony, who scored an injury-time equaliser for Swansea against QPR on New Year's Day, is on Africa Cup of Nations duty with Ivory Coast for at least the next three weeks.
But the player's agent Francis Kacou said last Wednesday that the deal should be completed soon.
Man City were without injured strikers Sergio Aguero and Edin Dzeko for the festive period and manager Manuel Pellegrini is keen to add further depth to his squad.
The Chilean, 61, had indicated he was not looking to do serious business in the January transfer window.
The club were hit with Uefa sanctions last year for breaching financial fair play rules which included a spending cap of £49m, plus any money raised from transfer fees.
They spent around £48m in the summer on Eliaquim Mangala, midfielder Fernando, and goalkeeper Willy Caballero, although sold Jack Rodwell to Sunderland for a reported £10m and midfielder Javi Garcia joined Zenit St Petersburg for £13m.
But with Arsenal, Liverpool and Tottenham reported to be interested in Bony, it is the Premier League champions who have moved quickest to bolster their attacking options.
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