Il Corriere dello Sport says Roma are in the market for a new defensive signing and Godfrey is emerging as a serious target.
Godfrey has struggled for minutes this season under manager Sean Dyche, but is regarded highly by Roma sports chief Tiago Pinto.
Pinto will attempt to find "the best solution" to get a deal done.
Godfrey could arrive to replace Chris Smalling, who has disappointed coach Jose Mourinho over his fitness issues.
