The 23-year-old will arrive in January from SuperSport United, subject to a work permit being granted, having signed a pre-contract agreement. The transfer fee is thought to be around £1.5m.
Tottenham’s rearguard has been beset by injury with Jonathan Woodgate and Michael Dawson both long-term absentees, while Ledley King and William Gallas have also spent significant periods on the treatment table.
Khumalo had a trial with Spurs earlier this season and impressed Redknapp, who has been forced to field 11 different centre-back combinations already this term.
Redknapp said: “He’s not expensive and we had him over on trial and we like him.
“He’s got potential and we’ve got a tie-up with the club SuperSport so we’ve taken him on.
“It will be a great chance for the kid. He’s a great lad. He comes from a poor background and he’s desperate for a chance.”
Khumalo started all three of his nation’s matches at this summer’s World Cup and scored in the famous 2-1 victory over France.
After the World Cup, Khumalo, born in Swaziland, said: “I have always wanted to play in England and obviously in the Premier League, and it would be a dream to move to Tottenham.
“Playing in the Champions League would be another dream.”
The combative centre-half captained SuperSport United to a record-equalling third straight Premier Soccer League championship, having become the youngest captain to lift the trophy last year at the age of 22.