Tavernier, 26, joined the 'Gers' from Wigan Athletic in 2015 and has since featured over 140 times for the Scottish side, currently managed by Liverpool legend Steven Gerrard.
West Brom have made seven new additions to the side this summer, including Newcastle forward Dwight Gayle and Manchester City defender Tosin Adarabioyo, both on season-long loan deals.
According to 'Sky Sports', West Brom have also rejected a bid from Watford to take 28-year-old defender Craig Dawson on loan with a view to a permanent deal.
West Brom continue their Championship campaign on Saturday at Carrow Road against Norwich City.