Bonucci is regarded as a key member of the Juventus side.
The report says that City boss Pep Guardiola has always liked the 29-year-old Italy international. They report that Manchester City were prepared to pay the defender 7 million Euros (£5.98 million) per year.
However, the Serie A champions actually view him as an important part of their side, hence them turning down the offer, but there is a hint in the report that City could return with an improved offer.
It seems clear that Guardiola is intent on adding to his defence this summer, with Everton's John Stones also strongly linked to the club, including by the Daily Mail.
City certainly appeared to struggle at the back for much of last season, especially when captain Vincent Kompany was absent from the side, as the Belgian struggled with a series of injuries last season.
The former Barcelona and Bayern Munich manager has made a series of signings already ahead of his first season in charge at the Etihad Stadium and he seems intent on improving the spine of the team.
Bonucci recently impressed for Italy during their Euro 2016 campaign, where he shone not only in his defensive duties, but on the ball, in particular with his passing range for Emanuele Giaccherini's goal against Belgium. Such a quality is one that would fit nicely in a Guardiola side.