The Republic of Ireland interantional only moved to the KC Stadium in January for £7.5m from West Brom and scored four goals in 15 games for Steve Bruce’s side.
Saints’ new manager Ronald Koeman wants the 27-year-old to form a strike partnership with Italian Graziano Pelle, who followed the Dutchman from Feyenoord.
Koeman made the former Reading forward his number one target after Manchester United’s Javier Hernandez apparently rejected a move to the south coast club.
The Mexico international has been told he can leave United, but is keen to move to a club playing in Europe.
Long‘s possible arrival at St Mary’s follows the departures of Adam Lallana, Rickie Lambert, Luke Shaw, Dejan Lovren, Dani Osvaldo and Callum Chambers.