The France international now has just six weeks remaining on his contract, making it likely that he will depart Parc des Princes on a free transfer.
Although PSG will remain in contact with the World Cup winner in a last-gasp bid to agree fresh terms, Los Blancos seemingly regard themselves as favourites at the present moment.
According to Le Parisien, Real officials have been telling people that the 23-year-old will soon complete the move to La Liga.
A different report has previously said that Mbappe will not make any announcement on his future until the end of the Ligue 1 campaign, which concludes on May 21.
Mbappe has contributed 30 goals and 19 assists in his 41 outings in France's top flight and the Champions League this season.