The Times report that City are considering a move for the Brazilian as they weigh up their transfer options this summer.
Pep Guardiola has recently signed Kyle Walker but the club are known to still be in the market for full-backs.
Danilo is valued at around £21 million by the Spanish giants. The Brazilian was unable to hold down a regular spot at Madrid under Zinedine Zidane, making just 17 league appearances last season.
Reports have linked City with AS Monaco defender Benjamin Mendy as well as the aforementioned Danilo.
Chelsea will still feel confident about completing a deal for the Brazilian as they also eye a move for Alex Sandro to kick start their summer recruitment drive.
The Premier League champions confirmed the arrival of Tiemoue Bakayoko from Monaco yesterday, as Antonio Conte’s side continues to take shape.