The England winger is a subject of interest for the Spanish giants after missing the giant capture of Kylian Mbappe from PSG.
The fee is said to be around £50million, with Los Blancos looking to lure the 27-year-old away from Premier League winners in the final 12 months of his contract.
Sterling has struggled to hold down a place in Pep Guardiola’s champion side this season but has still produced 17 goals and nine assists in 47 appearances.
In an interview with Spanish media last year, Sterling admitted his strong admiration for Los Blancos even when pictured with the club’s shirt draped over his shoulders.
“It’s one of those things… At the moment I’m at City and I’m really happy. But I say Real Madrid are a fantastic club, when you see the white shirt you know exactly what the club stands for. , it’s huge. But at the same time, I have a contract with City now and I have to respect it. But it’s a fantastic club.
Since joining the Citizens from Liverpool in 2015 in a controversial £49million move, Sterling has developed into a world-class striker, despite his apparent lack of skill in front of goal.
He scored 131 goals in 339 appearances for City, putting him only behind Sergio Aguero in the clubs’ all-time leading scorers list.
If the deal is done, it will likely see Manchester City head into the futures market despite already signing Erling Haaland from Borussia Dortmund earlier this month.