Navas' future has come under regular scrutiny this season and his desire to remain with the European champions was not aided by an error in the second leg of Madrid's eventual Champions League quarter-final win over Juventus.
Madrid were close to signing promising Spain shot-stopper Kepa Arrizabalaga from fellow La Liga outfit Athletic Bilbao in January, while they are known to be admirers of Manchester United's David de Gea.
Perez, though, is a fan of Navas, whose standing among the club's fans was stamped by his induction as an honorary member of the Ramon Mendoza Supporters' Group at their 30th-anniversary event.
"Keylor, you have won the love and affection of everyone in Madrid," Perez said in quotes reported by AS.
"You have done that through talent, professionalism and by your conduct, because you have identified with Real Madrid's values.
"We are very proud to have you as our goalkeeper, regardless of what others might say. Keep working hard and I'm sure you will be with us for many years to come."
Navas has two seasons remaining on the six-year deal he signed upon joining Madrid from Levante in 2014.