Kylian Mbappe was reportedly the subject of text messages sent to Real Madrid chiefs by club president Florentino Perez after his sensational Champions League display on Tuesday.
The Paris Saint-Germain striker came off the bench to score a hat-trick as Thomas Tuchel's side secured a comfortable 5-0 victory away at Club Brugge.
It came just a few weeks after Real had to rely on second-half goals from Sergio Ramos and Casemiro to salvage a draw against the Belgian side on home soil.
Mbappe, 20, has long been heavily linked with a move to Spanish giants and AS claims that his performance in Belgium prompted a number of WhatsApp messages between Perez and other high-ranking officials at the Bernabeu about a potential transfer for the Frenchman.
According to the Spanish publication, they read: "Have you seen Mbappe's performance? In the second half he has scored a hat-trick against the same Brugge side that had us on the ropes at the Bernabeu.
"This is a signing that would get the fans on their feet and put the club on top for at least ten years."
Tuesday's victory in Belgium saw PSG stay on top of Group A after taking a maximum of nine points from their first three games of the competition.
The French giants return to domestic action this weekend when they welcome Marseille to the Parc des Princes, looking to extend their lead at the top of Ligue 1.
Meanwhile, Real picked up their first win of this season's Champions League at Galatasaray on Tuesday after the nervy draw with Brugge and a heavy defeat in Paris.
Germany midfielder Toni Kroos scored the only goal of the game in the first half in Istanbul to ease the pressure on manager Zinedine Zidane and put Real second in the group with four points from their first four games.
