Spain international Busquets clashed with Bilbao striker Aritz Aduriz in the second half of the season opener at the San Mames and sustained a knock to his right ankle.
Brazil's Alves has a suspected adductor muscle strain after tangling with the same player in the first half, Barca added on their website.
Both had to come off and were replaced by Sergi Roberto and Marc Bartra respectively.
"This is a contact sport and we are not having any luck, although the injuries don't appear to be serious," coach Luis Enrique told a news conference.
"The pre-season meant we had to start competing very early and there is a greater risk of things like this happening," added the former Spain and Barca midfielder.
Barca have already contested two trophies this month, beating Sevilla 5-4 to win the European Super Cup and losing 5-1 on aggregate to Bilbao in the Spanish Super Cup.
(Reporting by Iain Rogers, editing by Mark Meadows)
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