The Evening Standard's James Robson noted Los Blancos remain keen on Sterling, whose contract will expire in the summer of 2020.
Robson believes Sterling's two-goal effort in England's 3-2 UEFA Nations League win over Spain on Monday, coupled with Real's interest, should encourage the Citizens to make fast work of offering the 23-year-old a new deal.
Sterling's coming-of-age game at the international level is "compelling evidence" the player can thrive for Real, according to Robson. Los Merengues have Sterling's name on a list of forwards that includes Chelsea's Eden Hazard and Paris Saint-Germain's Kylian Mbappe.
Rumours of Real's fondness for Sterling have been building. ESPN FC's Jonathan Smith reported how La Liga giants Real "are watching developments," while Sterling remains no closer to signing a new contract.
The idea of a Sterling move to the Santiago Bernabeu isn't fanciful, considering how much the wide forward has kicked on in recent seasons. Always known for his pace, Sterling has added varied movement and a ruthless streak in front of goal.
Those qualities not only recommend him to Europe's top clubs, but they also make fighting to keep a player with his best years ahead of him an easy decision for City:
Sterling's improvement for the Three Lions is mirroring the strides he's made under Pep Guardiola in Manchester. The former Liverpool ace has already hit four goals and provided a pair of assists through seven games in the Premier League this season, which is further proof of his rapid maturation into a consistent match-winner at the highest level.
City's problem will be finding a way to pay Sterling on a par with the many other creative talents in Guardiola's squad. Robson wrote it's unlikely Sterling will be given a deal to match the £400,000 per week handed to Kevin De Bruyne, but the former still "wants a substantial increase" on the £175,000 he earns.
While De Bruyne's vision and flair are vital to City's stylish brand of play, Sterling's direct runs and eye for goal offer a cutting edge Guardiola's men would sorely miss.
City know the focus Sterling and his agent, Aidy Ward, can bring to negotiations, having dealt with the pair when acquiring the player from Anfield in 2015. Sterling is in an equally strong position should clubs come calling to take him away from the Etihad Stadium.