The Saints leapfrogged the Hammers into sixth place in the Premier League with a fine 2-1 win away at Tottenham Hotspur at the weekend while Slaven Bilic's side were thrashed 4-1 at home by Swansea City.
It means West Ham must beat Manchester United in their game in hand and then match Southampton's result against Crystal Palace when they travel to Stoke City on the final day.
But should they fail to catch Ronald Koeman's side the ramifications could also be felt in the transfer market.
Saints to trump Hammers for Zaza?
And crucially Calciomercato reports that Zaza will only join a club that is competing in European competition.
The 24-year-old has been heavily linked with a departure from Turin after struggling for regular game time this season.
West Ham made an unsuccessful bid for Zaza last summer but Juve refused to allow him to leave having only just signed him.
Zaza has scored five goals in 17 appearances for the Bianconeri this season but is struggling for starts with the likes of Paulo Dybala, Mario Mandzukic and Alvaro Morata ahead of him in the pecking order.
West Ham do have the lure of London and their new 60,000 seater Olympic Stadium home, though.