Corriere dello Sport says Real Madrid, Juventus and Inter Milan have joined the battle for the Brazilian, who has all but accepted his future now lies away from the capital.
Olympique Lyon and Manchester City have confirmed their interest, however Juve president Giovanni Cobolli Gigli has also made contact this week. Claudio Ranieri, the Juventus coach, sees Mancini as an ideal successor for veteran Pavel Nedved.
Inter, meanwhile, see Mancini as a direct replacement for veteran Luis Figo.
Real coach Bernd Schuster has also instructed his board to compete for the Brazilian, admitting: "We need to improve our wing play."
Gilmar Veloz, Mancini's agent, is in Rome this week for crunch talks with directors over his client's future.