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.
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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)
