The centre-back's current contract expires in the summer of 2014 but there are hopes he will sign a new long-term deal at the Britannia Stadium.
Shawcross has been at Stoke since 2008 after graduating from United's Academy and has become one of their star players, which has attracted some Premier League rivals.
But Stoke chairman Peter Coates wants to prolong the defender's stay with the club and told The Sentinel: "Ryan is an important player for us.
"And we want to keep him and make him secure. You try to look after your assets.
"He has developed his career at this football club and we want him to stay, so it's up to us to find a way of keeping him."