The Premier League champions believe the deal represents excellent value for a 19-year-old with just six months left on his contract and who has still to make a league start for the club.
Diaz will be given a chance to prove himself in the second half of the season, and will be presented at 13:30 on Monday following a medical.
City will receive 15 per cent of any future transfer fee, or 40 per cent should neighbours Manchester United buy him from Real Madrid.
The Spanish midfielder is leaving Manchester City for Real today in a deal rising to €20m.
MANU FERNANDEZ APReal Madrid soccer player Brahim Diaz
The signing of Diaz is the latest example of Madrid's aggressive transfer policy in recent years which has seen them hoover up young Spanish talent with varying success.
He moved to the Premier League from Malaga in 2013 but has struggled to establish himself in City's first team.
Despite his lack of senior action City boss Pep Guardiola was keen to keep hold of the player and the club made several lucrative offers to retain his services, but were unable to convince him that he will play more in first-team football than he will in Madrid.
In Diaz's absence, Riyad Mahrez, Raheem Sterling and Gabriel Jesus comprise a strong forward line for City, with his fellow academy graduate Phil Foden in midfield alongside Kevin De Bruyne and Ilkay Gundogan.