Real were held to a goalless draw by Athletic Bilbao on 2 December despite creating numerous opportunities, with Cristiano Ronaldo and Karim Benzema drawing a blank again in front of goal.
The result meant the Madrid club are still eight points adrift of Barcelona, who had drawn 2-2 with Celta Vigo earlier on the same day.
Adding to Zidane's troubles, Sergio Ramos was sent off for two bookable offences at San Mames, while Dani Carvajal and Casemiro both picked up their fifth bookings of the season, meaning the trio will be suspended for the crucial clash against Sevilla next weekend.
A defeat to Sevilla would push Real, who have only won three of their last seven matches, out of the top four in La Liga and considerably raise the heat on Zidane.
Speaking at an event honouring Real's longest-serving members, Perez said Zidane is the right man to continue delivering success to the club.
"We have a team to keep dreaming with a wonderful trainer in Zinedine Zidane who has achieved everything with Real Madrid, the team of his life," he was quoted as saying by AS.
"He is one of our great symbols, one of the biggest icons who brings together the values and talent necessary to be the leader of one of the best squads we have ever had, captained by Sergio Ramos and Marcelo. They, too, are examples of our values."
Zidane signed a new three-year deal at Real in August, extending his commitment to the club until 2020.
The 45-year-old is the fourth-most successful coach in Real Madrid's history with seven major title wins, including one La Liga title and two Champions League crowns.
He took over as Real coach in January 2016 on a two-and-a-half-year deal following the dismissal of Rafael Benitez.