Perri, 28, arrived at Elland Road last summer from Lyon and signed a contract running until 2029, making 16 Premier League appearances last season and keeping three clean sheets as Leeds finished 14th in their first campaign back in the top flight.
His potential departure comes as Leeds themselves push to reshape their goalkeeping department, having opened formal talks with Manchester City over a move for England international James Trafford, whose game time at the Etihad has been severely restricted since Gianluigi Donnarumma's arrival from Paris Saint-Germain last September.
Trafford, whose City contract runs until 2030, returned to the club from Burnley last year but has found himself behind Donnarumma in the pecking order, with Daniel Farke identifying the 23-year-old as his priority target following the departures of Illan Meslier to Arsenal and Karl Darlow to Manchester United.
Leeds United are understood to be leading the race for his signature ahead of Newcastle, who had previously shown long-standing interest, and Aston Villa, who also evaluated his availability earlier in the window.
Torino, who finished 12th in Serie A last season, view Perri as a potential upgrade in goal as the Turin club look to strengthen ahead of the new campaign. Should Leeds lose Perri and successfully complete a deal for Trafford, it would represent a significant reshuffle in goal for Farke's side within the space of a single transfer window.
For now, talks between Leeds and City over Trafford are understood to be progressing, with Perri's own situation likely to become clearer in the coming days as Torino push to finalise their interest before the transfer window closes at the start of September. Leeds still have back-up options in Alex Cairns should Perri depart, though the club's recruitment team are determined to bring in a proven senior starter to secure the position for the long term.