The 19-year-old has only just broken into the Forest first-team this season, but has done enough already to catch the eye of the Potters and Toffees, who are both said to be preparing bids.
Forest are desperate to keep hold but it has been suggested they may have to cash in as a takeover involving US investors continues to drag on.
The youngster is not tied to a long-term contract either, which puts Forest in a vulnerable situation heading towards the January transfer window.
They were forced to cash in on star winger Oliver Burke in the summer and the same could happen to Worrall, with Stoke waiting to pounce if given any encouragement.
Southampton have announced plans for a two-week tour of China as part of their pre-season preparations for the 2018-19 season.