Cajuste spent last season on-loan at Ipswich Town, where he caught the eye in a team that was ultimately relegated.
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Upon returning to Napoli, the midfielder has been transfer-listed this summer.
Sky Italia says he has now attracted an approach from Premier League new-boys Burnley. The move is being driven by Clarets manager Scott Parker.
Also interested in Cajuste is Ole Gunnar Solskjaer's Besiktas.