The Daily Star says Mancini has made De Rossi one of his top transfer targets and was desperate to make him his next big-name signing.
The Blues were willing to hand the Italian international a bumper £160,000-a-week deal in the hope of luring him to the Etihad Stadium.
Fresh negotiations between Roma director Franco Baldini and De Rossi's agent Sergio Berti have resulted in the 28-year-old agreeing a new five-year deal worth around £100,000 a week.
The contract will still include a release clause of around £20m, allowing De Rossi to join another club if they meet that amount.
De Rossi is concerned Roma might not secure Champions League football.
And failure to do so could allow Mancini to renew his interest in the World Cup winner in the summer.