Centre-back Shawcross has this week been linked with a move away from the Britannia Stadium to Barclays Premier League champions Manchester City.
But Scholes, quoted by The Sentinel, said: "It comes as no surprise to me what I have read about Ryan
I don't know if there is any truth in it, but certainly we have had no contact.
"Players like him and (Stoke 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."
Shawcross, 26, was named Stoke's player of the year at the end of a 2013-14 season in which the club achieved their highest Premier League finish of ninth and a points tally of 50.
He has been a consistent performer for the Potters since their promotion to the division in 2008, but did not make Roy Hodgson's England squad for this summer's World Cup in Brazil.
And Scholes added: "Ryan's had six great years in the Premier League and that's quite something for a player of his age.
"It is interesting that he has reportedly attracted the attention of big clubs in the way he didn't seem to attract Roy Hodgson's attention.
"Anyone around these parts will think he should be on that flight to Brazil with England."
Meanwhile, it has been reported that Stoke midfielder Steven N'Zonzi has handed the club a transfer request, as he did last summer.
And on that matter, Scholes said: "I wish I could give you a clear answer.
"It's been in the papers over the last week or so that Steven's father has been in touch with us and he's mentioned that Steven's ambition is to play at the highest level that he can play.
"He's alluded to the fact that maybe he would want to move on and do that.
"But I am being a bit circumspect because until I sit down with Steven and his father and hear what they've got to say and why they are saying it, I'm guessing too much."
Source : PA
Source: PA