The 26-year-old defender, who earned a solitary England cap in 2012, played all but one of the Potters' Premier League matches this season and helped propel them to ninth, their highest-ever finish in the division.
Shawcross has been cited as a possible target for clubs higher up the ladder, a development which does not shock Scholes.
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"I don't know if there is any truth in it but certainly we have had no contact.
"Players like him and [goalkeeper] Asmir Begovic, just to name a couple, will attract attention. That's part and parcel of being a successful football team.
"That's the way it's always going to be. It's one of the prices of doing well but it doesn't mean you necessarily have to do business with them.
"We don't need to sell. We've been going through our situation for next year, our budgets, what is available in terms of buying players and players' wages and we are not in a position where we need to sell.
"We are looking to strengthen and move forward."