The club are unhappy at the prospect of the 25-year-old running down the last 18 months of his deal.
Manchester City's Roberto Mancini says he likes him, and could be tempted to make a move for the former Manchester United man.
Shawcross, who made his England debut in November, has earned rave reviews this season as Tony Pulis' side have gatecrashed the top half of the Premier League.