The Dutchman, who joined Barca from Ajax Amsterdam in the close season for 13 million euros (£11.73 million), will miss La Liga fixtures against Deportivo La Coruna, Valencia and Granada plus two Champions League group games with Manchester City.
Cillessen is back-up to Marc-Andre ter Stegen, his only appearance this season coming in a shock 2-1 home defeat by Alaves.
Barca added that right back Sergi Roberto, who missed Spain's World Cup qualifying win away to Albania on Sunday, has strained the adductor in his right leg but is expected to be fit for Saturday's home game against Deportivo.
His Spain team mater Jordi Alba is set to miss the Deportivo game with a hamstring injury suffered in Thursday's 1-1 World Cup qualifying draw with Italy but the left back is expected to return for the visit of Manchester City on Oct. 19.
(Reporting by Richard Martin; editing by Ken Ferris)
