Real did not say how long the Portugal international, who has formed a solid partnership with Spain's Sergio Ramos for Real this season, would be out of action but local media reported it would be around two weeks.
Real play city rivals Atletico Madrid in the May 24 showpiece in Lisbon and have two more La Liga matches before then, Sunday's game at Celta Vigo and a home meeting with Espanyol on the final day of the season.
Atletico are top on 88 points, three ahead of defending champions Barcelona, with Real a point further back in third.
If either Pepe or Ramos are out, Real coach Carlo Ancelotti typically deploys France defender Raphael Varane, whose appearances have been limited this season following a serious knee injury.
(Reporting by Iain Rogers, editing by Justin Palmer)
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