Antonio Conte are desperate to add a new striker to provide competition for Alvaro Morata, who is without a goal in his last six appearances.
Dzeko is in his third season at Roma after joining from Manchester City and finished last season as the top scorer in Serie A with 29 goals.
The SUN UK quoted a source at the Stamford Bridge outfit as saying that the Blues have agreed a basic fee of £28 million with Roma to sign the 31-year-old.
However, negotiations are still ongoing over Dzeko’s full transfer fee including add-ons but it’s expected that the deal will cost in excess of £30m.
According to the report, Chelsea are also close to securing the signing of Dzeko’s Roma teammate, Emerson Palmieri.
The Brazilian full-back will compete with Marcos Alonso on Chelsea’s left flank.
Speaking on Sunday evening, Roma’s sporting director, Monchi admitted that Dzeko could leave this month.
He said, “Whatever happens will be done thinking about what’s best for the club and the team.
“In my mind is always the desire to strengthen the squad, which is my job.”