The injury also forced White to be replaced in Scotland's victory over Italy a couple of weeks ago. Tommy Hayes could also return to the squad for Friday's game after recovering from a knock.
Warriors captain Andy Nicol said: "Jason's return would be a major bonus for us. Although he hasn't taken contact, the medics are happy with the progress of his recovery. Tommy's prospects are less clear but he might just make it." Nicol is assisting caretaker coach, Richie Dixon, for Glasgow's remaining fixtures. He hopes to return in the No.9 jersey against Borders at Hughenden a fortnight on Friday for what will be his last appearance before he retires.
Glasgow must win both matches to have any chance of qualification for the Heineken Cup next season but Borders will be determined to avoid another season playing in the Parker Pen competition.