Spurs supremo Redknapp had revealed that a deal with Manchester United might have already been agreed for the diminutive playmaker, who is set to leave Chelsea when his contract expires on July 1, before expressing Tottenham's interest in his former West Ham charge.
"I've got a feeling that it might have been done [for Cole to join United]," Redknapp told talkSPORT.
"It wouldn't surprise me in the least. Joe's not wanted to speak to anybody and his people have been saying 'let's get the World Cup out of the way first.' But we'd be interested for sure. I like Joe."
Cole was thought to be at the centre of a three club tug of war between Manchester United, Spurs and Arsenal, but it now appears as if Cole will continue to play his football in the capital.
Tottenham now look set to rival North London rivals Arsenal for the player as talks continue tentatively with Cole's representatives.
Emirates Stadium chief Arsene Wenger is a known admirer of Cole's qualities, previously stating: “I like him as a player because when he played against us he always did a lot of damage against Arsenal.
“He can create impact through his creative play. He can dribble, he can pass, he can score goals.”
Cole is thought to be demanding at least £90,000-per-week in wages, money that may be freed up should Cesc Fabregas make his expected transfer to Barcelona after the World Cup.
Cole would become the fourth player in recent times to transfer between Arsenal and Chelsea, behind Ashley Cole, Lassana Diarra and William Gallas.
The 28-year-old midfielder missed most of last season through injury, but forced his way back into Fabio Capello's plans for South Africa after a dazzling display in the warm-up friendly with Japan.
Cole has already expressed his desire to decide on his future after the World Cup. He failed to agree on a new deal at Chelsea after making close to 200 appearances for the Stamford Bridge side.
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