The 28-year-old is entering the final year of his contract with Tottenham and could be on his way out of the London-based club this summer.
Tottenham are open to parting ways with the right-back and could sell him for around €12m, which is significantly less than the €25m they paid for his signature in 2017.
Aurier has been heavily linked with a return to PSG, with Mauricio Pochettino said to be keen to reunite with the Ivory Coast international in Paris.
However, PSG could face competition for Aurier, with La Liga giants Real Madrid also interested in acquiring the services of the defender, according to French outlet Foot Mercato.
Los Blancos have enquired about the possibility of signing the former Toulouse full-back, who has made 27 appearances across all competitions for Spurs this season.
Aurier is also said to have discussed the possibility of plying his trade in Spain with Real Madrid star and former Lens team-mate Raphael Varane.
A transfer to Real Madrid could be an exciting option for Aurier, but it remains to be seen if he is prepared to function as the understudy to Dani Carvajal.