Diaz, who moved to City in 2014, will be out of contract at the end of the season.
Real Madrid are keen to secure the signing of Manchester City youngster Brahim Diaz as soon as possible and are hopeful that a €10m bid will convince Pep Guardiola's side to part with the Spaniard in January.
Diaz, who moved to City in 2014, will be out of contract at the end of the season, meaning he would be available on a free transfer once he becomes a free agent. He has rejected several offers of a new contract at the club and is said to have his sights set on a move to Madrid.
News of Real's bid for Diaz comes from Spanish outlet Marca, who claim that City have now accepted they will lose the 19-year-old at the end of the season, and could find themselves unable to refuse a £9m bid for a player who would be available for free in just six months.
Diaz is reportedly desperate to move to the Santiago Bernabeu as soon as possible, and the two clubs are working hard to find an amicable solution to the situation.
City executive Ferran Soriano is said to have a close relationship with Real official Jose Angel Sanchez, and the pair are leading negotiations to finalise Diaz's return to Spain.
Ahead of City's Champions League clash with Hoffenheim on Wednesday, Guardiola appeared to admit that Diaz will leave the club, comparing the situation to Jadon Sancho's move to Borussia Dortmund in 2017.
Guardiola is quoted by Goal as saying: "It would have been our desire for Jadon to stay here, Brahim to stay here. That was our wish."
This season, Diaz has made three appearances in cup competitions for City, netting two goals in the Carabao Cup victory over Fulham in early November. He has made a total of 14 appearances for the club since joining from Malaga, but has often been forced to settle for short substitute appearances.
He made five appearances in the Premier League last season, but his longest outing lasted just 16 minutes. He has struggled to establish himself ahead of the likes of Leroy Sane and Raheem Sterling, and Diaz is said to believe that first-team opportunities will be easier to come by in Madrid.