The Saints had scouts at Marseille’s match against Lille, with N’Koudou reportedly their target.
The 20-year-old winger has been one of the stars of Ligue 1 this season, and he is in demand.
Tottenham Hotspur, Liverpool and Manchester City are all also reportedly keen on the youngster.
However, Southampton are believed to be keen to win the race, and they have been tracking the Frenchman’s progress in recent weeks.
N’Koudou has scored nine times in 24 matches this campaign, setting up four goals in the process.
If he did move to Southampton he would considerably strengthen manager Ronald Koeman’s attacking options, which have already been bolstered due to the addition of Charlie Austin earlier this month.
Southampton are currently eighth in the Premier League table with 33 points from 23 matches, but Austin's immediate impact could see them make a steady climb in the weeks to come and N'Koudou's addition would only bolster that aim.