Sunderland have made signing a striker their top priority this summer.
Sunderland are desperate to add a new front man to their group, to help support Jermain Defoe.
Defoe was Sunderland’s top scorer by a distance last season, and Allardyce wants the veteran front man to have more competition.
Sunderland could even switch to a 4-4-2 formation, should they secure a player who could work in partnership with Defoe.
The Black Cats have made Dzeko one of their top targets, but doing a deal could be difficult.
Dzeko is currently on £140,000-a-week at Roma, and he would have to take a wage cut to move to the Stadium of Light.
Sunderland do not want to break their wage structure, as they are concerned about the harmony of the squad, if Dzeko came in as their top earner.
The Black Cats are also said be slightly worried about spending £11 million on a 30-year-old who would have little resale value if he was to join.