Iturbe caught the eye while on loan at Verona from Porto last season, scoring eight goals in 33 top-flight appearances.
Verona subsequently signed the 21-year-old Argentine to a permanent deal for €15 million in May, but speculation over his future has been rife, with Juventus and Milan believed to have been among those in the running to secure his services.
Last month Verona coach Andrea Mandorlini conceded that Iturbe was likely to leave the club, and it is Roma who have swooped.
The deal acts a further statement of intent from Roma, who have already added Ashley Cole, Urby Emanuelson, Seydou Keita and Salih Ucan to their ranks as Rudi Garcia's men bid to go one better than their second-place finish last term.
The fee could climb to €24.5 million with performance-based add-ons.
