Osbourne faces at least six weeks on the sidelines but the club are unwilling to put a timeframe on his recovery for fear of putting pressure on the 26-year-old.
The former Coventry player suffered the injury during Aberdeen's 1-0 victory over Dundee on Friday night after returning to the team in January following a near two-month lay-off with a hamstring injury.
Manager Craig Brown told the club's official website: "To lose Isaac again last Saturday to another injury is a huge disappointment for the squad.
"Having just returned to the team in recent weeks, he had made a positive impact on our performances and he will be sadly missed.
"Isaac is a consummate professional and he will work extremely hard to ensure his rehabilitation from this injury is swift."