The Brazilian goalkeeper has reportedly chosen the Serie A side despite late interest from Olympique de Marseille.
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Torino’s pursuit of Perri had stalled after Marseille entered the race, but the French club’s attempts to hijack the deal have now failed.
The 28-year-old is keen to move to Turin after a difficult spell in West Yorkshire.
According to Sky Sports, Perri’s departure will leave Leeds searching for another goalkeeper.
However, any new arrival is expected to provide competition and cover for James Trafford, who joined the Premier League club from Manchester City in a record transfer earlier this summer.