Kessie, 25, is unable to sign contracts at the moment as he is with Ivory Coast at the Africa Cup of Nations.
But Sportitalia's Gianluigi Longari claims Kessie's representatives have verbally agreed to join Barca at the end of his Milan contract in July.
That appears to rule out Spurs' hopes of signing him this month after reportedly negotiating a fee of around £11million.
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