Madrid have drawn two of their opening four La Liga matches this season while on Wednesday evening they were beaten 3-0 by Paris Saint-Germain in the Champions League.
To make matters worse for Zidane, his marquee summer signing Eden Hazard struggled to make an impact while Madrid failed to register a shot on target against PSG.
According to The Sun, Perez is willing to give Mourinho a second stint at the Santiago Bernabeu and the Madrid president feels that the former Manchester United boss is the only man to get the club back on an even keel.
Mourinho, meanwhile, is open to returning to Madrid and felt he was a strong contender to get the job after Santiago Solari was sacked in March.
But several players in the Madrid dressing room, including captain Sergio Ramos, were against the Portuguese coming back to the Bernabeu and Perez opted to re-appoint Zidane instead.
Mourinho spent three years at Madrid during his first spell and won the La Liga title in 2012.