Stoke had been trying to sign Arnautovic to a contract extension, but the player was reportedly prepared to play out the final year of his deal.
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The huge 27-year-old Austrian has been with Stoke since 2013, and broke through with 11 Premier League goals last season.
Arnautovic would give Everton support behind Romelu Lukaku, and insurance in case the Toffees do wind up selling their Belgian striker. Everton also has Arouna Kone as a potential target forward, but Oumar Niasse is expecting to leave after less than a year at Goodison Park.