The 29-year-old was taken from the field on a stretcher in the Scottish champions' 1-1 pre-season draw with Rapid Vienna on Sunday.
Brown was missing from Celtic's 5-2 friendly victory over Linz on Tuesday, and manager Ronny Deila confirmed that scans had confirmed the Scotland international is set for a spell on the sidelines.
"It is not positive news for Scott. He will be out for two months, maybe three," he said after the Linz rout.
"That's hard for him and it's hard for us. But we can't do anything about it. We have to try to get him back on the pitch as quick as possible.
"The hamstring has been almost ripped off. Scott doesn't need an operation, which is a positive thing. However, it is quite a big injury."
Despite Brown's absence, Deila suggested he will retain faith in his squad rather than enter the transfer market, adding: "We have good midfielders who can step up while Scott is out."
Celtic have one more run-out against Dukla Praha on Friday before facing KR Reykjavik in the first leg of their UEFA Champions League second-round qualifier next Tuesday.
