The Los Blancos are keen to bolster their attacking ranks after losing Cristiano Ronaldo to Juventus in the summer.
And the Mirror claim Sterling is their new No1 priority, as they value the 23-year-old's potential.
Real Madrid have been interested in landing Chelsea star Eden Hazard for the past two years.
The Belgian, 27, even confirmed once again he still "dreams" of playing for the Galacticos – with Alvaro Morata stating the Blues are powerless to cling onto him.
Sterling has only 18 months left on his current £180,000-a-week contract at the Etihad and is yet to agree terms on a new deal.
The impasse has certainly attracted the attention of Real chief Julen Lopetegui – who is keeping tabs on the ex-Liverpool star's current situation.
Sterling starred for City last season as Pep Guardiola's men stormed to the Premier League title.
In 47 games for his club last season, Sterling scored 23 times, with his end product noticeably improving under the Spanish boss.
However, he drew a blank in 11 matches for England, including all six World Cup games, as Gareth Southgate and Co reached the semi-finals.
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