The Croatia international is set to leave the San Siro upon the expiration of his contract this summer, with no extension agreed between the two parties at this stage.
Having fallen out of favour, the 25-year-old could struggle to find his way back into Antonio Conte’s side for the rest of the season.
He joined Spurs from Real Madrid in 2020, and has a £27.5m buy-back clause for him to return to the Spanish capital, adding a rival edge to any interest from the Blaugrana.
The 22-year-old, who is contracted to Lille until the summer of 2025, was heavily linked with a move to Newcastle United in the January transfer window, but a move did not materialise.
Spurs are being tipped to raid the Italian market following Antonio Conte’s arrival.
With Kessie’s Milan deal due to expire at the end of the season and no renewal in sight, with that said, an intense battle for the Ivorian’s signature is expected to begin in 2022.