The midfielder, who has one year left on his contract at St Mary's, travelled from Nairobi to London on Sunday to undergo a medical and finalise a deal to become Mauricio Pochettino's first signing of the summer transfer window.
The 24-year-old will return to Kenya following an official unveiling, which is expected early this week.
The move will see Wanyama reunite with his former manager at Southampton, with Pochettino bringing the Kenya captain to English football from Celtic in a deal worth in the region of £12.5m in 2013.
Wanyama will compete with first-choice centre-midfield pairing Eric Dier and Mousa Dembele, both currently with England and Belgium at Euro 2016, as Tottenham embark on their first Champions League campaign in six years and look to build on their third-place finish in the Premier League.
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"It's always a tough moment leaving a club where you are so much used to their tradition and fans but it's always the dream of any footballer to play at the very top level like the Champions League," Wanyama reportedly told Soka25east on Saturday.
"I will definitely miss the awesome Southampton fans who always showed me love and I know they will fully understand my decision."
Wanyama's arrival prompts further questions over the futures of midfielders Nabil Bentaleb and Ryan Mason, who both struggled for minutes in midfield under Pochettino last season.
The pair, who broke into the first team after time spent with the youth setup, have been linked with summer moves away from White Hart Lane.