The Wales international attracted an enquiry in January from Bournemouth and is now on Everton's radar, says the Daily Mail.
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Having just turned 27, Thomas is enjoying career best form this season after working his way up the football pyramid after being released by West Ham as a teenager. He counts on Boreham Wood and Huddersfield Town among former clubs.
The winger is in Wales' squad this week for their World Cup playoff tie with Bosnia and Herzegovina.
Thomas' contract with Stoke runs to 2028.