The 26-year-old and goalkeeper Asmir Begovic have been touted as possible departures, but Scholes has claimed that the club are in a position to strengthen, not sell.
Premier League champions Manchester City have been reported as potential suitors for Shawcross, who racked up 43 appearances for the Potters in the 2013-14 season.
"It comes as no surprise to me what I have read about Ryan," The Sentinel quotes Scholes as saying. "I don't know if there is any truth in it, but certainly we have had no contact.
"Players like him and Asmir Begovic, just to name a couple, will attract attention. That's part and parcel of being a successful football team.
"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."
Stoke finished ninth in the league table.