The 22-year-old joined Lille in January 2015 and has impressed since being at the club.
The Morocco international has played 51 times for the Ligue 1 side and scored 15 goals.
Boufal can play either as a No.10 or as a winger and won the best African player in Ligue 1 award last season.
However, should Southampton sign the player, Saints fans will have to wait until mid-September to see the forward in action as he is currently recovering from a knee injury suffered in May.
It is thought that the likely arrival of the Moroccan at St. Mary's could trigger a loan move for Jay Rodriguez to West Brom, who is in need of game time after suffering two serious knee injuries.