"Zidane could do it. He has the qualifications and fulfils all the requirements but we have not thought about it yet," Perez told Spanish television station Telecinco late on Tuesday.
"It is something we have to consider, but as a first step it is a bit... I don't want to say risky but he needs to become hardened," Perez said of the Frenchman, who passed his coaching exams this year but has no experience in the job.
Perez, who was formally re-elected as president for a new four-year term on Sunday, recently confirmed that the former Madrid playmaker would have a central role to play at the club in the coming years.
"He could also lead the project from an office or the bench," he said. But to do both at the same time "is more complicated."
Real Madrid's first choice to take Mourinho's spot still appears to be Paris Saint-Germain coach Carlo Ancelotti, although Perez refused to be drawn too far on the matter so as to avoid upsetting PSG.
"Ancelotti is with a club that is a friend and a club that we must respect. From tomorrow we will work to see what the possibilities are. We don't want to do anything aggressive with any club," he said.