According to The Athletic's Sam Lee, Los Blancos are interested in Sterling and invited his camp for talks with director general Jose Angel Sanchez to lay the groundwork for a move in the coming years.
Sterling would welcome a transfer there, but he "knows he is in the best place for his game right now and ideally would not think about a transfer for another two or three years."
The 24-year-old has four years remaining on the contract he signed with City in November last year, but the Sky Blues' director of football, Txiki Begiristain, is said to have given him assurances the club will offer him a new deal by the end of this season.
City arguably have more strength in depth than any side in Europe, as they demonstrated on Saturday when they demolished Watford 8-0 without needing to bring on the Englishman, who was rested on the bench.
However, he remains one of the side's most important players having hit new heights under manager Pep Guardiola in recent seasons.
Last season, the winger notched 25 goals and 18 assists in 51 matches in all competitions, up from 23 and 17 in 46 games in the campaign prior. This season, he has six goals in seven games.
With those kinds of numbers, he wouldn't look out of place at a club like Real.
He's also become a force to be reckoned with on the international stage. After a three-year goal drought with England, he has now scored eight goals in his last eight Three Lions appearances, as well as producing six assists in that time.
Here's a look at a couple from England's 5-3 win over Kosovo earlier in September:
Sterling's rise has invited comparison with Cristiano Ronaldo and Lionel Messi, though Guardiola does not believe it is helpful to name him alongside them:
As outstanding as Sterling has been, such comparisons don't hold up to much scrutiny. Messi, for example, played one fewer game than Sterling last season but scored more than double the number of goals and made four more assists.
Nevertheless, the City star's consistently excellent form over the last two campaigns puts him among the best players in the world and one the club will be desperate to keep.
According to Lee, if Real continue to struggle as they have this season—they've drawn two of their first five games in La Liga and were beaten 3-0 by Paris Saint-Germain in the UEFA Champions League—it could accelerate their pursuit of Sterling to next summer.
The Sky Blues will be in a strong position if that comes to pass, though, all the more so if Sterling signs a new contract in the coming months.