Alan Pardew is currently in charge at the Hawthorns but he seems set to depart at the end of the season, if not before, unless he manages to lead them to safety.
Survival looks unlikely for Albion as they are eight points adrift of safety and the club are expected to begin the hunt for a new boss in the summer.
Smith has impressed during his time with Brentford and they remain in the promotion hunt in the Championship.
The Bees boss is unlikely to quit for West Brom if he does manage to take Brentford up this season but if they remain in the Championship the Baggies could launch a move, although it remains to be seen as to whether or not Smith would be tempted by the chance to replace Pardew.
Newcastle have been linked with a shock move for former Sunderland goalkeeper Simon Mignolet during the summer.