The pair are former Roma teammates and De Rossi knows Dzeko is having a frustrating time with Fiorentina.
Tuttosport says De Rossi has been messaging the veteran centre-forward about joining him at Genoa next month.
Dzeko has been stuck on the bench at the Viola and has managed just two goals this season. As such, he is ready to leave Florence in January.
And De Rossi is working on Dzeko about a short-term deal to the end of the season. However, the Bosnian will have to accept a cut on his Fiorentina wages, as they are out of range of Genoa's budget.