Cillessen starred in Barca's comeback win against Sevilla in the quarterfinal in midweek, making a stunning save from Andre Silva and keeping out an Ever Banega penalty.
As Barca's cup goalkeeper, he was set to start the first leg of the Clasico tie against Madrid at Camp Nou on Wednesday.
However, an injury picked up in training on Friday has ruled him out of that game and the second leg at the Santiago Bernabeu at the end of February.
Marc-Andre ter Stegen, who usually only plays in La Liga and the Champions League, is likely to replace the Dutchman in both games.
B team goalkeeper Inaki Pena, meanwhile, has been called up to take Cillessen's place in Barca's squad for Saturday's home game against Valencia in La Liga.
Cillessen joined Barca from Ajax in 2016 but his appearances have been limited due to the presence of German stopper Ter Stegen.
Despite that, he has endeared himself to the club's supporters with a number of fine performances in the Copa del Rey -- a competition he's won twice since joining Barca.
His display against Sevilla, as Barcelona came from 2-0 down to win 6-3 on aggregate, was perhaps his best yet.
A first half save from Silva and a penalty stop from Banega immediately after was hugely appreciated by the fans, who bellowed his name out for the majority of the first half.