The 26-year-old defender, who earned a solitary England cap in 2012, played all but one of Stoke's Premier League matches this season, only missing the penultimate game against Fulham through suspension following a red card in the 1-0 defeat to Tottenham.
Shawcross' impressive displays at centre-back helped propel Stoke to ninth - their highest Premier League finish - and speculation has mounted regarding a possible move away from the Britannia Stadium for the former Manchester United trainee.
However, Scholes stated on Thursday that Stoke had received no offers from other clubs.
"It comes as no surprise to me what I have read about Ryan," Scholes told the Stoke Sentinel.
"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."
