The price is less than the £12.5m Saints paid Celtic in 2013 and Tottenham have been able to drive a hard bargain because Wanyama has only one year left on his contract.
Wanyama, who joined Southampton when they were still managed by Pochettino, is thought to have been due for a medical on Monday as Spurs looked to tie up their first signing of the close season.
The 24-year-old, who is expected to sign a four-year deal, will add further muscle to the north Londoners’ midfield and depth to their squad as they prepare for a return to the Champions League next term.
Wanyama was sent off three times last season in 30 Premier League games – more than any other top flight player. He also received three red cards in 91 games for Celtic.
Spurs are also believed to be in the market for a striker to lessen the load on Harry Kane, and have considered a move for Holland And AZ Alkmaar’s Vincent Janssen.
Wanyama’s departure is a further blow for Southampton, who have already lost manager Ronald Koeman to Everton this summer. The Toffees paid £5m to release the Dutchman from his Saints contract.