The People says De Rossi, who penned a new five-year deal worth almost ¬£8million-a-year at Roma in February, is Mancini's prime midfield transfer target.
Mancini is desperate to get De Rossi on board but must win over football chief Brian Marwood and the rest of the City hierarchy.
He warned: "It would be a mistake to think that because we won one championship we are at the top.
"We want to stay at the top for a long time. But now is the time to defend it for the first time."