The 22-year-old enters the final year of his contract with the Ligue 1 club at the end of the season and Real have made no secret of their admiration for him.
Goal reports that the La Liga side will make Mbappe their top target for the next transfer window.
And the outlet has revealed that the France international has started house hunting in the Spanish capital.
The youngster has employed a representative to find a house in Madrid, but it’s unclear if this is a move to simply acquire property in the city, or a hint of a future move.
Mbappe hasn’t made a firm decision on his future yet and intends to wait for the end of the season before considering his next move.
PSG are still in the running for the Ligue 1 and Champions League titles, and they face Manchester City in the semi-finals of the latter competition next week.
When asked about his future recently, Mbappe said: "I am in a period of reflection, but if I sign with PSG it is to be there for many years."
The World Cup winner has been in superb form for Mauricio Pochettino’s side this season, scoring 35 goals in 41 appearances in all competitions.
That included a run of eight goals in his last five Champions League matches.
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