The 28-year-old joined the Baggies last summer from Arsenal and made 33 Premier League appearances but could not prevent the club from suffering relegation.
Torino, meanwhile, have secured ninth-place finishes in Serie A over the past two seasons, and previously had fellow England international Joe Hart on loan at the Stadio Olimpico.
According to esteemed Italian journalist Gianluca Di Marzio's website, the Granata have already made initial contact regarding the transfer of the former Gunners full-back.
The report also claims that Torino are currently testing the waters and could push for a loan deal with an obligation to buy the following summer.
Gibbs came through the ranks at Arsenal and made 230 senior appearances for the North Londoners between 2006 and 2018.