Reports have even gone as far as suggesting that Real will be the first to know should Chelsea decide to cash in on the France international at some point.
That looks highly unlikely as Kante is a crucial player for the Blues, but reports claim that Real and Chelsea came to an agreement over the summer, whereby Los Blancos will get first refusal on him, should he move on.
Real snapped up Eden Hazard from Chelsea over the summer and it is at that point where the agreement over Kante is also said to have been reached.
The Spanish giants are keen to bolster the heart of their midfield and Manchester United star Paul Pogba is a top target for their coach Zinedine Zidane, however the same can be said of Kante.
Zidane is a big admirer of his fellow countryman and there was talk of them trying to do a deal during the recent transfer window.
However, even though Chelsea were willing to do business over Hazard, there was no chance of Kante leaving as a replacement could not be signed due the club's current transfer ban.
Good news for Chelsea is the fact that Kante has said that he is very happy at Chelsea and is not interested in moving on despite interest from Real, Juventus and Paris Saint-Germain.
Chelsea are also in a strong position as Kante still has four years remaining on his contract so there is no rush to get him tied down to a new one.
Kante is valued in the region of £100million which would not be a problem for Real, but any bid would only come at the end of the season as they spent heavy during the last window and the pot is empty for now.