Dzeko is up for sale at Roma after falling out with coach Paulo Fonseca. The Bosnian clashed with Fonseca over the sacking of team manager Gianluca Gombar earlier this month.
Gazzetta dello Sport says United have been offered the chance to sign Dzeko.
Odion Ighalo has left the club following the end of his loan spell from Shanghai Shenhua.
Both Manchester clubs have been offered the chance to sign Dzeko before the deadline falls next week. The veteran enjoyed great success with City before his move to Serie A.