The 25-year-old Ivory Coast international, who moved to the French Ligue 1 side from Portuguese outfit Pacos de Ferreira in 2015, has enjoyed an impressive season with six goals and nine assists to help Nice up to third in the league under the guidance of coach Lucien Favre.
Reports claim Saints boss Claude Puel is keen to raid his former club and take Seri to St Mary's in the summer, where the pair would be reunited having spent last season together with the French outfit.
The main obstacle is that a host of top clubs around Europe have been alerted by Seri's good form this season and will almost certainly look to secure his signature in the transfer window.
In addition to the Saints, Premier League rivals Leicester and German outfit Borussia Dortmund have also been mentioned as potential suitors.
Nice are believed to be looking to pick up 40 million euros for Seri which could prove to be prohibitive for the Saints.