The Sun say the Potters have contacted Old Trafford chiefs regarding the winger's availability.
Stoke need to source a replacement for Marko Arnautovic.
The Austrian ace look set to join West Ham for £24m - and Mark Hughes wants Young to replace him.
Young was heavily linked with a move to the Chinese Super League last season.
But Jose Mourinho was keen to keep the experienced campaigner on his books.
Young is yet to feature on United’s pre-season tour of the United States.
He could face Manchester City in Maryland early on Friday morning, though.
The former England international has been deployed primarily in a full-back role over the last two seasons.
However, a move to Stoke could see him utilised further up the pitch.
The Sun say other clubs are also sniffing around - although they don't specify who.
Young is one of United's longest serving players having joined the club from Aston Villa for £17m in 2011.