The midfielder came back for Saturday's match against Kilmarnock after missing four games with a knee injury but was hurt again towards the end of the game.
"It could be eight to 12 weeks," Deila told a news conference ahead of Thursday's Europa League Group A game against Ajax Amsterdam at Parkhead.
The club do not yet know whether Brown will need surgery but said his current problem is not related to his previous knee injury, he added.
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(Reporting by Steve Tongue; editing by Ken Ferris)
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