According to Mark McAdam at Sky Sports, Austin has passed his medical and has agreed terms with the Baggies.
The transfer fee is reportedly worth around £4m, seeing Saints recoup what they paid QPR for him in 2016.
McAdam says Austin’s contract will be a two-year deal with the option of another year.
This move would mark a comedown for Austin, who reportedly rejected a move to West Brom in July.
West Brom are struggling for firepower after selling Salomon Rondon and Jay Rodriguez this summer.