Spanish sports daily Marca claimed on its front cover that Madrid club captains Casillas and Ramos had issued the ultimatum to Perez in a lunch meeting on Tuesday.
Perez took the unusual step of responding directly to the speculation, branding the report an 'attempt to destabilise the club' in a press conference convened at short notice at the Bernabeu.
A short time later a joint statement from Casillas and Ramos was posted on the club's official website, in which the
The statement read: "The two captains of Real Madrid want to communicate the following: 1. That the meeting held on Tuesday with the president and CEO of the club never raised any ultimatum from us regarding the character of our coach.
"2. As captains of the staff and members of an excellent group of people and professionals, we categorically reject the use of our names and the rest of our squad's in support of information that is entirely false.
"3. We want to show our support for the character of our coach,
"4. Finally, we wish to convey to the members and fans of Real Madrid our unalterable conviction that only through the unity of everyone at Real Madrid will we overcome all the challenges that we face each season.
Club president Perez was unequivocal in his criticism of the article in his press conference held in the directors' box at the Bernabeu.
"It's totally false that there was un ultimatum from the players," Perez said.
"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 have only come here to deny a news story that has destabilised our club and the players.
"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."
After winning La Liga at a canter last season, Mourinho has come under scrutiny this term after presiding over an indifferent run of results that has left the club trailing league leader Barcelona by 15 points.
Reports of in-fighting between the outspoken coach, his playing staff and several club directors have never been far from the surface, while the Portuguese courted controversy recently by dropping talismanic club captain Casillas from successive league games.
There have been high points this season, with Madrid safely into the last 16 of the UEFA Champions League and into the semi-finals of the Copa del Rey after completing a 3-1 aggregate win over Valencia.
Speaking before the Valencia match, Mourinho again conceded the title race was beyond Madrid and said the focus must now be on success in the cup competitions.
"The league title is impossible now - there is too big a gap," he told RMC Sport.
"We have to focus on other goals like the Copa del Rey and the Champions League."
Mourinho signed a contract extension to 2016 in the wake of Madrid's league triumph but he has continually been linked with a move away from the club, with a return to
The controversy came as it was confirmed that goalkeeper Casillas faces a long spell on the sidelines after fracturing a bone in his hand during the cup clash with Valencia.
Casillas took a kick to the hand after 18 minutes of the 1-1 draw, which safely saw Madrid through to the semi-finals.
A club statement confirmed Casillas had a fracture.
"After carrying out initial X-rays at the
"It will be assessed again by a specialist later today to decide what treatment will be followed."
While no estimate was given on how long Casillas might be sidelined, an injury of this type could take up to three months to heal.
That will mean a return to duty for back-up