According to the Daily Mirror, the 18-year-old fullback has been watched by City and Chelsea scouts, and most recently Everton boss David Moyes' dispatched coach Jimmy Lumsden to assess the Championship player.
The England under-19 international came though the Barnsley academy, before making the step up to the first-team towards the end of last season.
Stones has already made six appearances this year, and the 6ft 2in defender has impressed in a Barnsley side that's begun the season in fine form.
Chelsea and Manchester City are reportedly interested in bringing him to the Premier League, but they now face competition from Everton.
The Toffees have also enjoyed an impressive start to the season, and David Moyes is thought to be keen to secure a long-term replacement for Tony Hibbert.
Seamus Coleman now operates further forward at Goodison Park.
Following interest from Roberto Mancini and Roberto Di Matteo, Barnsley issued a 'hands-off' warning to ward off interest from Chelsea and Manchester City, but they would likely be brought to the negotiating table if the offer was big enough.