Spurs had an £18 million offer turned down for Wanyama last summer, according to the Mail, which reported earlier this month that the Londoners had revived their interest ahead of the forthcoming transfer window.
The Kenyan has only a year to run on his Saints contract and has stalled on the club's offer of a new one, making a summer exit extremely likely.
Both Spurs and Leicester are keen to add to their squads ahead of their respective Champions League campaigns, though the Foxes arguably have little need for Wanyama if Kante remains at the club.
However, the Mirror reports the Premier League champions-elect are set to lose the France midfielder, with Arsenal, Manchester City, Chelsea and Paris Saint-Germain all keen.
Tottenham could find themselves a little light in the middle of the park this summer, with Tom Carroll out of contract and the future of Nabil Bentaleb uncertain.
Should both players depart that would leave them with only Eric Dier, Dele Alli, Mousa Dembele and Ryan Mason as senior central midfielders.