Pickford became the third-most expensive goalkeeper in football history on Thursday when he swapped Sunderland for Everton – his transfer fee only eclipsed by new Manchester City signing Ederson and Juventus legend Gianluigi Buffon.
Butland is only one year older than Pickford but has already made his senior international debut with England and enjoyed more Premier League experience.
And should Stoke be inclined to sell him – which they have no intention of at present – they believe any offers would now have to exceed the one that secured Pickford's services.
"Jack's a very special talent," said Scholes. "He's done extremely well for us and has done well for England.
"I think most people objectively would argue he's some way beyond Jordan Pickford, I think, but that's not what we are thinking about.
"We are thinking about what Jack can do for our football team and Jack is a key part of what we want to be next year.
"He's a player we missed last year, albeit Lee Grant came in and was magnificent and deservedly won the player-of-the-year award.
"But when you've got a player of Jack's quality you want him available for your team."
Stoke had to cope without their number one goalkeeper for the majority of the 2016-17 campaign due to an ankle injury, but he returned in the closing weeks as Mark Hughes' side finished 13th.