The Bosnian is understood to have agreed personal terms with the Serie A club and is now expected to travel to Italy later on Wednesday to conclude the deal before undergoing a medical early on Thursday.
Dzeko moved to the Etihad Stadium from Wolfsburg four years ago for £27m and has scored 72 goals in 187 appearances for the club.
But he has not always started – with almost half of his outings coming from the bench – and now looks set to continue his career in Italy.
Dzeko only scored six club goals last season and City are willing to let him leave after he fell down the pecking order in Manuel Pellegrini’s plans.
The Chilean coach has already shipped out Stevan Jovetic to Inter Milan on loan and has the prospect of pairing Sergio Aguero with £49m signing Raheem Sterling in attack.
City begin their Premier League campaign at West Bromwich Albion on Monday, a game you can see live on Sky Sports 1 HD.