Schneiderlin has told the Saints he wants to leave the club and has been attracting plenty of interest, particularly from Arsenal and Napoli.
The Gunners have been monitoring the Frenchman's situation for some time and could now be tempted into making an official approach with news that he wants out of St Mary's.
However, Serie A club Napoli has already held talks with the 24-year-old's representatives and settled on a deal to raise his annual salary by 400,00 euros, taking it to nearly €1 million plus bonuses.
Rafa Benitez's side also believes it is battling Tottenham for Schneiderlin's signature, and would ironcially strengthen Spurs' need for a central midfielder if the club is successful in its attempts to sign Sandro.
Schneiderlin could join up with former Saints boss Mauricio Pochettino should he decide to move to Spurs.
The Argentine boss moved to White Hart Lane at the end of May, sparking a firesale at the south-coast club.
Shaw completed his switch to Manchester United on Friday, while Lallana joined Liverpool.
Striker Jay Rodriguez, who had been tipped to make England's World Cup squad until damaging his medial knee ligament towards the end of the Premier League season, has also been linked with a move to Tottenham, while center back Dejan Lovren is said to be on Arsenal's radar.