Wanyama is set to undergo a medical at Spurs on Monday after the two clubs agreed a fee over the weekend.
Press Association Sport understands Tottenham will pay around £9million up front but the deal could reach almost £12m if additional clauses are met.
Wanyama would become the first Saints player to be reunited with the club's former manager Mauricio Pochettino at White Hart Lane.
Spurs expressed interest in the 24-year-old last summer but Saints held firm and were rewarded, as Wanyama played a key role in leading the team to a sixth-placed finish.
The Kenyan, who has only one year remaining on his contract at St Mary's, has made 85 league appearances for the south coast club, scoring four goals.
Pochettino is eager to boost his options in midfield with Tottenham hoping to balance Champions League football with another title challenge next season.
The Argentinian is also eager to bring an extra forward to provide cover for Harry Kane but Spurs are yet to follow up talks with AZ Alkmaar striker Vincent Janssen.
Janssen visited Tottenham's training ground two weeks ago and spoke to the club about his future but Tottenham have not submitted an official bid.
The Dutch striker is currently on holiday in Barcelona and has also held talks with German club Wolfsburg, although a move to Spurs remains the 22-year-old's preferred choice.
Source : PA
Source: PA