Brown suffered the injury during Saturday's pre-season clash with city rivals the New York Jets, collapsing on the floor untouched after an interception return.
The 26-year-old led the Giants in interceptions last season with eight, and coach Tom Coughlin admitted that his side "definitely" would miss him.
"He made a lot of plays for us. So we'll definitely miss him," Coughlin is quoted as saying by ESPN.
The Giants are also currently without starting safety Antrel Rolle, who is hoping to have recovered from a sprained ankle in time for the regular season opener.