Asensio became a Madrid player in 2014, but the Spaniard had loan spells at Mallorca and Espanyol before returning to the capital for the start of the 2016-17 campaign.
The 21-year-old scored 10 times in all competitions for Los Blancos last term, including three in eight Champions League appearances as the Spanish giants retained the European Cup.
Asensio also starred for Spain Under-21s at the Under-21 European Championships earlier this summer, with the attacker inspiring his team to the final of the competition.
According to Don Balon, PSG are firm admirers of the attacker's talents, and the French giants are prepared to pay £130m in order to convince Real Madrid to part with the Spaniard in this summer's transfer window.
Asensio started 11 of Real Madrid's 38 La Liga matches last term.