The press conference came days after Barcelona's 2-0 El Clásico victory confirmed them as La Liga champions for a second consecutive season, completing another trophyless year for Real Madrid.
Pérez, 79, who has been president since 2009 having also served between 2000 and 2006, addressed claims in the Spanish press that he lacks the energy for the role — and did so with considerable force.
"I'm here to say I'm not going anywhere and I'm here to stand for election. If someone does want to stand for election, this is your opportunity. But don't move in the shadows with publications and news outlets."
He denied reports about his health and insisted he remains fully operational.
"I work hard. I'm up early and I'm the last to go to bed. I understand there's frustration out there. But if people have forgotten, we have won 37 titles since I've been in charge. I would like to know the people who are saying these things. In this board of directors, I want whoever is leaking this to come out."
He turned on La Liga at length, identifying the competition's governance as a structural enemy of the club.
"And of course, the enemy — La Liga. We have to battle against La Liga. Corruption in the game, systematic."
Despite the confrontational tone, he expressed confidence that the club's 100,000 members remain behind him.
"I lead the biggest club in the world and you wouldn't imagine the praise I get. But 100,000 members are behind me and are very happy."
Pérez declined to address the managerial situation beyond what was already public. José Mourinho is widely reported to be set to return to the club he managed between 2010 and 2013, but Pérez would not confirm anything. Arbeloa is expected to leave when his contract expires.
Barcelona's legal department responded to Pérez's comments about La Liga and the broader accusations with a statement confirming they were studying his words carefully and evaluating next steps.
Real Madrid will finish the season without a major trophy for the second consecutive year. Pérez's press conference was the defiant statement of a man who has no intention of going quietly — and who intends to appoint the next manager himself.