The La Liga giants are believed to have enquired after the Manchester City midfielder, who has proved the stand-out figure in the top-flight during the opening half to the campaign.
De Bruyne has scored five goals and provided ten assists, having played a key role in the Citizens' amazing run of 20 consecutive wins in all competitions, with Pep Guardiola's men sitting eight points clear at the top of the Premier League table.
The 26-year-old still has three years remaining on his contract but City have moved recently to open talks over a new deal which will reputedly make him the highest-paid player in the club's history on £290,000 per week.
Spanish outlets have stated Real are planning a revamp of their ageing star-studded squad next year and they do see De Bruyne as an attractive target, asking De Koster if he would be tempted by a big-money move to the Bernabeu.
The Belgium international became City's record buy when he was signed for £54.5m from Bundesliga side Wolfsburg back two seasons ago but the word coming back from people close to the player is that he is happy in Manchester and has no desire to push for a move away.