The former AC Milan midfielder has instructed agents to find himself a Serie A club this summer.
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Tuttosport says Kessie has been offered to Inter Milan and Juventus, with the Ivory Coast international seeking €5m-a-year to sign.
Inter chief Piero Ausilio is keen, where he has added Kessie's name to his midfield shortlist.
Making the deal attractive for Inter and other Serie A clubs is Kessie becoming off contract at the end of June this summer.