The 23-year-old Russian was dropped by Red Bull-owned Toro Rosso last season after he had previously lost his place at the Red Bull Racing senior team to young Dutch sensation Max Verstappen.
Kvyat had also been linked to former champions Williams, who have yet to announce a replacement for now-retired Brazilian Felipe Massa.
Ferrari have several young drivers on their books, including Monaco-born rookie and Formula Two champion Charles Leclerc who will make his debut for Sauber, and Italian Antonio Giovinazzi.
(Reporting by Alan Baldwin, editing by Ken Ferris)
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