Sullivan said he will stay in Pittsburgh to rehab the injury, and the team will evaluate his status when they return. The Penguins host the Ducks on Monday.
In addition, Sullivan said the Penguins will be without Eric Fehr for "probably a month" as he has a "significant" lower-body injury.
"It's going to be longer-term," Sullivan said. "It's not season-ending. It's nothing that severe. But it's significant enough that it's going to be longer-term."