The 20-year-old France international, who has burst into Real's team this season, fell awkwardly after a tackle in the 19th minute and had to be substituted.
"His meniscus is affected. We will see what the scans say in the morning," club director Emilio Butragueno told Canal Plus television.
In the post-match news conference, coach Jose Mourinho added: "I don't want to speak ahead of the doctors, but it looks relatively serious and it is likely he will miss the Cup final and the two or three remaining games we have."
Real play city rival Atletico Madrid in the Copa del Rey final at the Bernabeu on Friday, their last chance to secure major silverware this season, after the draw at Espanyol saw Barcelona crowned league champions.
Varane has been one of Real's outstanding performers in recent weeks but Mourinho sought to play down the significance of losing the player.
"We have alternatives. It's a position where we have people who are playing, and who are fit," he added.
"Obviously, we are sad that he could miss a game everyone would want to play in, but we aren't going to over-dramatise because we have options in that position."
Varane was playing at right back against Espanyol, but Real have Sergio Ramos, Ricardo Carvalho, Raul Albiol and out-of-favour Pepe as possible alternatives at centre-back.