The Portugal international, whose participation at next year's World Cup in South Africa is now in doubt, landed awkwardly after jumping for the ball with Valencia striker David Villa in Real's 3-2 win at the Mestalla.
He was carried off in obvious pain and limped away from the stadium on crutches before tests on Sunday showed he had damaged the anterior cruciate ligament.
"He will be operated on next week and the recuperation period will be around six months," doctor Juan Carlos Hernandez told the club's website on Sunday.
"It's touch and go and difficult for him to make it (to the World Cup), but the most important thing is that he recuperates well and the knee is stabilised," he added. "He has many years of football left in him."
The likely absence of Pepe's towering presence for the rest of this season could expose Real's defensive frailties.
He missed the start of the season through suspension and team mates including Raul Albiol, Ezequiel Garay and Christoph Metzelder have looked badly exposed at times this term.
Valdano said Real would assess the chances of finding a replacement.
"There is no other Pepe on the market," the Argentine said on the club's website.
"We will assess the possibility of compensating the squad so we can feel more secure but that's something to be discussed from Monday," he added.
"We don't rule it out but there is also no certainty about it. We'll have to look closely at the market but what is certain is that a high quality player of ours has been injured and that there is nobody like him in the world."
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