After allowing club-record goalscorer Cristiano Ronaldo to join Juventus after nine trophy-laden seasons, Real have actively pursued a high-profile attacker to replace the Portugal captain.
However, after being priced out of deals form Mohamed Salah and Harry Kane from Liverpool and Tottenham respectively, the Spanish giants renewed their interest in Sterling.
According to Sky Sports, the club is not expected to make an official bid before Friday's transfer deadline but will continue to monitor his progress with England and the Citizens until the next transfer window.
Last season the former Liverpool man scored 23 goals in 46 games across all competitions as City won the Premier League title with a record 100 points and Los Blancos have been enticed by his development under manager Pep Guardiola.
Sterling has less than two years on his current contract and has been dragging his feet on committing to the club, with whom he currently earns £175k a week.