And last time things were this bad, the Real Madrid president did just that.
Back in February 2006, Los Blancos were dumped out of the Copa del Rey by Real Zaragoza, lost to Real Mallorca to fall 10 points behind Barcelona in La Liga, and suffered a 1-0 defeat by Arsenal in the first leg of their Champions League last-16 tie.
“I hope this decision will help us change direction and I believe that we still have a chance of achieving something in the European Cup this season,” said Perez, stepping down after six years in charge.
Those three defeats took place over 12 days; this time around, Madrid suffered the same fate in less than a week.
Arch-rivals Barcelona inflicted the first two wounds, knocking Madrid out of the Copa del Rey on Wednesday and moving 12 points clear in La Liga with another win at the Santiago Bernabeu on Saturday.
Ajax twisted the knife as they romped to a 4-1 win on Tuesday (5-3 on aggregate), leaving Madrid facing nearly over five months without any meaningful matches.
For the team which has reigned over Europe for more than 1,000 days, winning the last three Champions Leagues, this situation is unbearable.
"I've never felt so bad; I don't know how to explain it,” whimpered defender Dani Carvajal after the Ajax elimination.
Despite the disappointing performances of some players, and their hapless coach Santiago Solari, the finger of blame can only be pointed at Perez himself.
Madrid fans were quick to do so, too, having seen the team’s quality level reduced year on year, with only Zinedine Zidane’s fine man management and legendary top scorer Cristiano Ronaldo maintaining their European dominance.
Both left in the summer and Perez was unable to replace either satisfactorily.
The president failed to appoint any of his first-choice options as manager, instead wrecking Spain’s World Cup tilt by persuading Julen Lopetegui to sign a contract with the club on the eve of the tournament in Russia.
Then, when Ronaldo joined Juventus for €100 million (£86/$113m), Madrid could have reinvested that money in a new striker, someone who could offer at least 30 of the 50 goals a season Ronaldo guaranteed every year.
But instead Perez brought back Mariano Diaz, who barely gets included in Madrid squads these days, and asked the existing attackers – Gareth Bale, Karim Benzema and Marco Asensio – to step up their production.
Luka Modric attested to that before the Ajax clash when asked about Madrid’s lack of goals this season.
“The club wanted the other players to make up for what he did, divided between the strikers and that is not easy,” explained the Croatian Ballon d’Or winner.
Perez was even blamed by Ronaldo himself for his departure.
"The president looked at me in a way that suggested I was no longer indispensable if you know what I mean,” the Portuguese said. “That is what made me think about leaving."
After sacking Lopetegui, Perez went for the easy option and promoted Solari from the B team. It seemed to work at first but handing the Argentine a contract until 2021 seemed like a bad idea even at the time – and now Madrid will have to pay him off to get a new coach.
Solari made it fairly clear that he wouldn’t be quitting of his own accord after the Ajax game: "I did not come to the club in such a difficult moment to give up." His departure is only a matter of time, though.
Furthermore, Zidane and Ronaldo not being effectively replaced is only the tip of the iceberg that has sunk Real this season. Perez's problems go way back.
Colombian playmaker James Rodriguez was the last big signing made by the president, who splashed out €80m (£69m/$90m) that summer on the World Cup golden boot winner.
Since then, Perez has focused on signing younger players, refusing to shell out for stars in their prime unless there is a good deal to be cut.
He has tightened the purse strings to focus on what he wants to be his legacy – the renovation of the Santiago Bernabeu. Natural, for a construction magnate, of course, but unfortunately for Madrid, it’s costing them on the pitch.
Finishing 17 points behind Barcelona last season reflected the real state Madrid are in, despite their Champions League final victory over Liverpool. Things have to change.
On a huge billboard outside the Bernabeu reads an advert from a mobile phone network telling prospective customers to “look for the love of your life”.
Inadvertently, it seems to be a subconscious message to Perez too, as talk of the return of his beloved Jose Mourinho gathers momentum.
A week of complete and utter collapse seems to be driving the Portuguese coach and his former club back together. This type of disaster makes you start to wonder if Mourinho’s return might actually be a good idea but, in reality, it is another sign that Madrid are in dire straights – and it is Perez who has taken them there.
Madrid fans want him to go – although elections are not due until 2021 – and strict rules he approved in 2012 make it difficult for anyone to challenge him: a club president must have been a member for the last 20 years and have a personal guarantee of over €75m (£64m/$85m) from a Spanish bank.
The rebuild Madrid must undertake for next season is one of the biggest challenges in the club’s history and Perez must either follow the fans’ cries to resign, or fully commit to it.
The last few years have seen him take half-measures when it comes to the first team, which should be the lifeblood of the club, and this week they are now counting the cost of his parsimonious approach.