But the two clubs are yet to settle on a fee for the Bosnia and Herzegovina international, who has won two Premier League titles in four seasons with City.
Dzeko started just 11 league games last season, and Manuel Pellegrini this month admitted the £27m signing’s future could lie away from Manchester.
City committed as much as £49m to signing Raheem Sterling this summer and also have January arrival Wilfried Bony ready to start his first full season at the club.
But Steven Jovetic is set to join Inter on loan, and Dzeko’s departure would leave just Bony and Sergio Aguero as recognised senior centre-forwards.
Roma have already raided the Premier League once this week to sign Arsenal goalkeeper Wojciech Szczesny on loan.