The 21-year old was outstanding in Albion's 2-1 win over Stoke City at the weekend and has attracted attention from the likes of Leicester City and Everton in recent months.
However, Southampton are looking to head off rival interest in the England U20 international with Albion's financial position meaning they are open to offers.
Fellows, who set up both Albion's goals on Saturday, impressed Southampton when he faced them at the end of last season and interest has remained.
Southampton are making a move for highly-rated West Brom forward Tom Fellows
Russell Martin wants to bolster his forward options and with money coming in from Charly Alcaraz going to Flamengo, they hope to add striker Yuri Alberto from Corinthians and Albion winger Fellows.
Meanwhile, Southampton may revive interest in Arsenal keeper Aaron Ramsdale as a move for Feyenoord's Justin Bijlow has faltered following his medical.
Ajax also remain keen on Ramsdale.