Their confidence stems from their willingness to offer Andros Townsend or Benjamin Stambouli as part of the deal, report the Evening Standard.
Chelsea also have the France international on their wish-list, but Spurs now believe he will come to White Hart Lane.
Chelsea currently have the midfielder as just one of their back-up options while they pursue big name targets like Koke and Paul Pogba, while Arsenal are having second thoughts due to the fast development of Francis Coquelin.
Tottenham tried to buy Schneiderlin last summer only for Southampton to block his departure.
But the 25-year-old, who enjoyed a fine relationship with Spurs manager Maurico Pochettino before he left St Mary's Stadium for north London in May 2014, wants to leave the club again this summer.
Southampton will demand over £20m for his signature, but Spurs are prepared to offer squad players Townsend or Stambouli to help convince them to sell and for a lower price.
Townsend has started just 10 League games this term, however can still command a large fee in the transfer market due to his involvement with England, where he scored a great goal against Italy last month.
Stambouli has had a bit part role since joining from Montpellier last year, but would be a ready made replacement for Schneiderlin as he also plays as a defensive midfielder.
Pochettino is also considering making another move for striker Jay Rodriguez, even though he has not played for over a year.
Rodriguez suffered a serious knee injury in Southampton's defeat at Manchester City 12 months ago.
The 25-year-old, who scored 17 goals last term, is due to finally make a comeback in their final games of the campaign next month.
Pochettino wants to make sure he is fully over his knee problems, but is determined to strengthen his strikeforce and add to Harry Kane's threat, especially as he plans to get rid of Emmanuel Adebayor and Roberto Soldado.
However, Southampton's Head of Football Development Les Reed claims the club will not listen to offers for any of their players this summer having sold Rickie Lambert, Luke Shaw, Adam Lallana, Dejan Lovren and Callum Chambers in 2014.
Reed said: "It wouldn't be our intention to sell any of our players this summer.
"We're in the process of renegotiating contracts with some but our main purpose at the moment is keeping this squad together and adding to it."
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