The extent of the injury was confirmed by Lobbe's club, Toulon, following Saturday's Top 14 defeat by Toulouse.
Lobbe had come on as a replacement in the second half but would last just 15 minutes. He will now miss Argentina's Test matches against England, Wales and Italy.
"He will see a doctor in a Marseille to have an operation" read a Toulon statement. "His unavailability is estimated at two to three months."
This would cast doubt over his availability for the Heineken Cup return leg against Glasgow Warriors at Scotstoun on January 17.