Sterling has enjoyed a stellar season so far, already scoring a career-high 13 goals across all competitions including nine in just 13 Premier League appearances to help City into an eight-point lead at the top of the table.
The 22-year-old has been particularly important late on in games, winning City nine extra points with goals in the 82nd minute or later this season, including three late winners in a row against Feyenoord, Huddersfield Town and Southampton recently.
The England international's form has led to speculation that City could offer him a bumper new £300,000-a-week deal, but his willingness to postpone contract talks has reportedly piqued the interest of European champions Real Madrid, who are understood to be planning a busy January transfer window in order to turn their season around.
Los Blancos, who currently sit eight points behind La Liga leaders Barcelona, are particularly looking to add to their attacking depth and see Sterling as a possible option for their front three.
However, Madrid have also been linked with Chelsea's Eden Hazard, who is a long-term target for president Florentino Perez, while Real Sociedad's Alvaro Odriozola and goalkeepers Thibaut Courtois and Kepa Arrizabalaga are also understood to be on their list of targets.