Italian transfer expert Gianluca di Marzio, of Sky Italia, reports the tests took place in Lugano (Switzerland) in recent days.
Off contract in June, Kessié will join Barca for free and sign for the next four seasons, until 2026, earning €7m-a-year.
The negotiations would have accelerated at the end of February once it became clear that Kessié would not accept the renewal offer from Milan.
The Ivory Coast midfielder is viewed at Barca as a long-term successor for Sergi Busquets.
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