Ronald Koeman has stated that the Kenyan is not for sale, however.
"I don’t know if he is really a target for Tottenham… There is no bid. He is not for sale," said the Dutchman.
An experienced central midfielder - along with an additional attacker - is seen as the priority on the transfer front for Spurs, before the close of the window.
While Eric Dier has impressed with his performances in a holding role, relying on him as first-choice for a whole campaign would be risky.
Wanyama would also appear to be the perfect candidate for the White Hart Lane club, given his Premier League experience and familiarity with Pochettino.
Securing his services will be no easy task, however, and the Saints will be reluctant to sanction the exit of another key man in a summer that has seen the sales of Nathaniel Clyne, Morgan Schneiderlin and the loss of Toby Alderweireld.