O Jogo says Coentrao has become a mainstay of the Real first team and his compatriot Ronaldo is being courted by affluent French side PSG.
Moreover, with the future of Jose Mourinho still uncertain, City want to capitalise and strike while the iron is hot.
And money appears to be object, as they are bankrolled by their super rich owner Sheikh Mansour.
According to the Portuguese newspaper, City are prepared to offer £134m for the pair.
The one barrier could be that Ronaldo may not to wish to go to the Etihad as he would incur the wrath of bitter rivals Manchester United.