AS goes so far as to suggest Benitez is more likely to become the 11th coach to be dismissed by Perez before he has the chance to win the club's 11th European Cup.
According to the Spanish newspapers, the players no longer trust his methods.
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Such claims have been rejected by the agent of the former Valencia, Liverpool, Inter and Chelsea coach.
"Ultimatum? We don't know anything about that, no-one has formally communicated anything," agent Garcia Quilon told Mediaset. "Everything is speculation."
Meanwhile, Marca says there are people close to Perez advising that Benitez should dismissed, arguing that this is already a lost season and something needs to change.
The paper also claims former player and youth coach Zinedine Zidane is on stand-by to succeed Benitez if Perez acts.
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"Nevertheless, Zizou is well aware that anything is possible in football, all the more so at Real Madrid," said a report in Marca.
"He knows that if the present situation drags on or gets any worse, the phone could very well ring at any time with the voice on the other side offering him the keys to the first team earlier than expected."
In the aftermath of the 4-0 drubbing by Barcelona at the Bernabeu last month, Perez said Benitez retained the full support of the club but both publications suggest that is no longer the case.
There’s a concern in Perez’s office that the anti-Perez chants that punctuated the Clasico defeat could return if Madrid struggle at home in the Champions League last 16 against Roma in February.
Perez could make a change before the meeting with the Italian side to prevent the uncertainty over Benitez affecting Madrid’s Champions League future.
AS also hints that the number of injuries suffered by Madrid under Benitez is as a result of the players not responding well to Benitez's training sessions.
That claim was refuted by his agent, who said: "He has a correct relationship, both on a sport and human level."
Marca note that Benitez is aware that he is losing support at the club and as such is becoming closer to his coaching staff. Benitez is also said to be unhappy at his treatment by the local press.
Perez for the first time began to question Benitez’s position after the first half in Villarreal, Marca continue, and AS say that defeat at El Madrigal has put Benitez in an "extreme situation".
Additional information from Football Espana - The ultimate website for English-speaking fans of Spanish football
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