The former United midfielder has tipped one of Louis van Gaal's current crop to be top dog at Old Trafford for the next 12 years.
The 19-year-old moved to Old Trafford from Southampton in a big-money switch this summer but his United career 'has not got going yet,' according to Scholes.
The former United midfielder does think, however, that Shaw has a big future ahead of him if he fulfils his potential.
To do so, the 39-year-old adds, Shaw must take advantage of his opportunity to play for the England Under-21s.
"A great example of a player who would benefit from playing in the Under-21s is Luke Shaw, who is in Southgate’s squad for tonight’s game," Scholes wrote in his column for the London Evening Standard.
"Shaw’s United career has not got going yet but if he fulfils his potential he could be the left-back at the club for the next 12 years and with England too.
"He would benefit in a big way from playing at another tournament, possibly even winning this one. To his credit, Shaw has said he wants to play next June, providing England qualify, and I hope United encourage him."