The 21-year-old’s current contract at the Parc des Princes expires in 2022 and PSG are keen on tying down their superstar to a new deal.
Talks over extending his stay in Paris have so far stalled and the French World Cup winner may end up being sold next summer.
Mbappe has been linked with Madrid and Liverpool, and Pires believes a move to the Bernabeu would be the best option to further his career.
The Frenchman told AS: ‘Of course. I’m French, I like Paris Saint-Germain and I’d like him to stay in Ligue 1.
‘But if he wants to keep growing as a player, he should sign for Real Madrid.’
Adil Rami, Mbappe’s former international team-mate, feels that the striker’s time at the French club is ‘done’ and he will definitely move on at the end of the season.
‘I think that it is done,’ Rami told RMC Sport.
‘Even more so because talks are failing to advance. You get the sense that it annoys him, talking about that.
‘His future seems totally traced out and it is sad for the Parisians, even for Ligue 1.’