The 19-year-old endured a difficult spell at Norwich earlier this season but has penned an extension to his Spurs contract prior to joining the Championship outfit.
Kane spent the first half of the season on loan at Chris Hughton's Norwich City, but after making his Premier League debut against West Ham in September, he was sidelined for six weeks with an ankle injury.
The 19-year-old Kane subsequently struggled to break back into the first team, featuring just six times, and he returned to Spurs in January.
Kane won Millwall's young player of the season award last campaign for his successful spell at The Den, during which he scored nine goals in 24 games. He has also represented England's Under-19s, scoring six goals in 14 games.
Tottenham, meanwhile, also announced that young goalkeeper Jordan Archer has signed a contract extension following an impressive loan spell at Wycombe, for whom he has made 21 appearances this season. Also 19, Archer has represented Scotland's Under-21s four times and enjoyed a successful spell on loan at Bishop's Stortford last year.