The latest reports emerge amid stories suggesting that the 24-year-old is the subject of interest from English top-flight rivals Tottenham Hotspur.
According to the Daily Mail, Spurs are primed to make a £16million offer for the Saints midfielder, in a move that would see him reunited with former club manager Mauricio Pochettino, who made the switch to White Hart Lane last summer.
The Argentine coach clearly sees the 24-year-old, who scored three goals and created 16 chances for the Saints during the previous domestic campaign, as the man to solve their problems in the midfield area.
Manager Ronald Koeman had previously gone on record affirming that there would be no more high-profile departures from the south coasters this summer, according to Sky Sports, but Wanyama's actions appear to have thrown a spanner in the works.
The Kenya international has made two appearances for the Saints so far this campaign but the latest developments could see him miss the club's upcoming clash with Watford, as the player and club look to resolve the obvious question marks regarding his future.