The Serie A giants are also keen to sign Southampton bad boy Dani Osvaldo, who is currently on loan to rivals Juventus.
The 2010 treble winners have endured a difficult season in Serie A, and currently sit in fifth place and are unlikely to qualify for next seasons Champions League.
The club have recently been taken over by Indonesian businessman Erick Thohir, who is keen for a major squad overhaul in the summer.
New strikers are thought to be the priority, with Inter’s current crop of Rodrigo Palacio, Mauro Icardi and Diego Milito having managed just 16 league goals between them.
Inter are understood to be long term admirers of City forward Dzeko, who has started just half of this season’s Premier League games under Manuel Pellegrini.
Meanwhile, the San Siro side are also looking to sign Southampton’s club record signing Osvaldo, who was shipped out to Juventus on loan for the second half of the season following a training ground bust up.
The 28-year-old earned his big money move to the south coast after two impressive seasons for Roma, and has plenty of experience in Italian football.