Mane was dropped to the bench and only made a 10-minute cameo having turned up late for a team meeting prior to the game - the second time that the Senegalese forward has committed that offence.
The likes of Manchester United, Chelsea, Paris Saint-Germain and Bayern Munich have all been linked with a move for Mane in the wake of the fall-out with Koeman, while Arsenal are understood to have a long-standing interest in Wanyama, who handed in a transfer request during the previous window.
However, PA reports that the Saints are determined to keep hold of both players as they look to climb back up the Premier League table and challenge for Europe this season.
Mane has scored three goals in 20 league appearances for the club this season, while Wanyama has started 18 times in the top flight during the current campaign.
Southampton are in FA Cup action against Crystal Palace on Saturday before hosting Watford on Wednesday in their next Premier League game.