The report appeared in Spanish newspaper Marca, but Perez has moved quickly to deny it as a "lie".
"It's totally false that there was un ultimatum from the players," Sky Sports News quotes Perez as saying.
"I have not spoken to Mourinho. I don't normally come out to deny reports, but this was too important. This crossed an ethical line.
"I understand some people want the coach or president out, but what has been published is a lie. We the board are the only ones that will decide the future of our club."
Mourinho has been in charge of the La Liga giants since 2010.