The France midfielder played the full 90 minutes in Southampton's 2-1 defeat at Anfield on Sunday but Spurs will not be deterred and remain hopeful of securing a deal.
Goal UK understands that Mauricio Pochettino remains determined to land his former player and has sanctioned a bid of around 21 million euros.
Spurs are confident that Schneiderlin still wants to make the move to White Hart Lane, despite playing the full 90 minutes against Liverpool on Sunday after it had initially been suggested that he would pull out of training and matches.
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However, Southampton is refusing to grant the Frenchman an easy exit from St. Mary's Stadium, with manager Ronald Koeman reiterating that he is not for sale.
Speaking after the 2-1 defeat at Anfield, where Schneiderlin was denied a late equalizer only by an outstanding Simon Mignolet save, Koeman explained that he had no concerns about the France international's commitment.
"He’s a player of Southampton," the Dutchman told reporters. "He wants to play for us — he changed his mind and his mentality last week. He was tired in the last 10 minutes but he is a good player and he is important for us."
Following a summer of big-name departures from the south coast that has brought over 100 million euros in transfer fees to the club, Southampton does not need to cash in on Schneiderlin and is reluctant to let another of its major assets go.
The player has claimed that he has a gentleman's agreement with the club to allow him to leave and Southampton's resolve could be tested by Spurs' final attempt to sign him.
Tottenham leads the race to capture Schneiderlin after the 24-year-old rejected an offer from Arsenal. Arsene Wenger refused to offer any guarantees of a starting role at the Emirates Stadium, something that is believed to be more likely in Pochettino's side.
Spurs could allow Sandro to leave White Hart Lane if they are successful in their pursuit of Schneiderlin, with Napoli among the Brazilian midfielder's admirers. Sandro was not in the squad that beat West Ham 1-0 on Saturday.