Alba started Sunday's 1-1 Liga draw at Real Sociedad but missed Monday's training session and tests confirmed a heavy bruise on his right knee and a right ankle strain.
"How he evolves will determine his availability for the next few games." said a Barca statement.
A hamstring injury has limited the Spain international to just six league starts this season.
Turkish midfielder Arda Turan missed Sunday's game with flu but has recovered.
Barca begin their King's Cup defence on Wednesday with a trip to Alicante to face Segunda Division side Hercules in the first leg of their last 32 tie.
(Reporting by Adriana Garcia; Editing by Ken Ferris)
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