"I am not going to resign," the 79-year-old Perez told a press conference towards the end of a season in which Real Madrid will finish trophyless.
Barcelona defeated Madrid 2-0 in El Clasico at Camp Nou on Sunday to claim back-to-back league titles with three games to spare.
Madrid bowed out of the Champions League with a quarter-final exit against Bayern Munich.
Current Real coach Alvaro Arbeloa, who replaced Xabi Alonso in January, is unlikely to remain at the helm next season. Former Madrid coach Jose Mourinho has been linked with a return.
This season has seen Madrid finish without a major trophy for a second year running.
Fans have been protesting star player Kylian Mbappe over a perceived lack of commitment, while there have been reports of multiple training ground bust-ups, including one that sent midfielder Fede Valverde to hospital.