Speaking on Spanish football chat show Punto Pelota on Monday night, Perez ruled out a bid for Falcao in the next transfer window in January but suggested Real could table one in the close season.
Falcao, 27, left Real’s city rivals Atletico to join Monaco at the end of last season for a fee media reported was around 60 million euros ($81.5 million).
“He (Falcao) won’t come in January but in June, who knows?” Perez said.
“Nothing is impossible and there is a lot of time between now and then,” the construction magnate added.
“Falcao is a great player and I am aware that he wants to play for (Real) Madrid. I am aware of that, but it’s normal. They told me.”
One reason Real may be in the market for another striker is the patchy form of their French international Karim Benzema.
A favourite of Perez, Benzema has failed to win over Real’s demanding fans since his arrival in 2009 and is regularly whistled by some sections of the Bernabeu crowd.
Perez sought to defend the 25-year-old, who has been criticised by Real coach Carlo Ancelotti for not working hard enough on the pitch.
“I have faith in all the players in our squad,” Perez said.
“Karim is a great player and we get offers for him all the time, every summer.
“All I can say is that the coach is the one who has the authority to do what he believes is right.”
Perez urged Real fans to get behind the players and suggested the whistling could destabilise the team.
“I know how demanding the fans are. The players at this club have come to run, make sacrifices and fight, that’s what our fathers taught us and that is what we teach our children but we have to moderate these demands.
“I am greatly disturbed by this whistling of our own players in our own stadium.
“I ask that our members moderate the whistles and the criticism of our players because, if not, it could weaken us.”
Perez also discussed the planned refurbishment of the Bernabeu stadium and raised the prospect of a sponsor’s name being added to the giant arena.
“We want to start the works at the Bernabeu in the summer,” he said.
“It will be emblematic, to create a Bernabeu for the 21st century. We are looking for a sponsor and although the stadium will definitely still be called Bernabeu it could also have the name of a sponsor.”
Spanish media have reported that the remodelling of the stadium will cost 400 million euros and the venue may take the name “Santiago Bernabeu Fly Emirates” after the club’s Dubai-based partner.
(Reporting by Iain Rogers; Editing by John O’Brien)