Ramsdale, 28, returned to Staplewood for the start of pre-season this month, having spent last campaign on loan at Premier League Newcastle United.
The former England shot-stopper arrived at Saints from Arsenal in 2024, with then-Russell Martin's side paying an initial £18million to sign him.
Ramsdale was one of the club's few good performers that season, which ultimately ended with relegation to the Championship on just 12 points.
Saints owner Dragan Solak admitted Ramsdale wanted to move back to the Premier League, and Newcastle swooped with a £4m loan fee to acquire him.
But they opted against making his signing permanent, leaving Ramsdale - whose wages are too big for second-tier Saints to carry - on the market.
Sky Italy report that Torino, who finished 12th in the top flight last season, are interested in signing the goalkeeper, who has two years left on his deal.
They detail interest in a loan transfer, while fellow Italian publication SportMediaSet add that Torino are 'all about' adding him to their squad.
The report adds that sporting director Gianluca Petrachi is working on a deal to sign Ramsdale, who played 23 times for Newcastle last season.
Saints are all-but certain to offload Ramsdale this summer, having signed number one goalkeeper Daniel Peretz on a permanent basis in June.