The Sunday Mirror claims the Italian was targeted by the club last summer, but they still retain an interest in him and are ready to make him another offer.
Monaco are currently in the second tier of French football under former Chelsea boss Claudio Ranieri, and are on course for a return to the top flight.
However, they have lofty ambitions under billionaire owner Dmitry Rybolovlev, and see Mancini as their best long-term option, the newspaper claims.