Cole is available on a free transfer after his contract with Chelsea was not renewed and he has several Premier League clubs on his trail.
His former club West Ham United, where he made his professional debut under Redknapp, have already expressed a wish to bring the 28-year-old back to Upton Park.
Redknapp has also repeatedly stated he is keen on the midfielder, but has previously cast doubt on his prospects of being able to bring Cole to Tottenham.
Earlier in the week, Redknapp suggested a deal may already have been done for Cole to join Manchester United, only for the Red Devils to reportedly rule out a deal for the playmaker.
And now Redknapp has stepped up his efforts to add Cole to his squad, although he accepts he may have to wait until after the World Cup finals before being able to speak to the player.
"We've declared our interest in Joe to his agent, but that is as far as it has gone," Redknapp told The Sun.
"We are told Joe doesn't want to talk to anyone about any move until after the World Cup is over."
Following United's apparent lack of interest in a deal, Spurs' North London rivals Arsenal are currently favourites to land Cole.