Sky Sports says Chelsea are only willing to cough up £26m for Dzeko.
The experienced forward wants around £90,000-a-week and a three-year contract.
The Blues have been linked with a huge selection of target men in recent days.
Reports earlier this week claimed Antonio Conte personally contacted Dzeko in a bid to lure him to Stamford Bridge.
Dzeko, who has bagged 61 goals in 117 appearances for Roma, wants to make sure he get enough game time in west London.