Cole is out of contract after leaving Chelsea and has been linked with a move to Manchester City, Liverpool, Arsenal and Spurs.
Redknapp revealed he would "love" to sign the 28-year-old England international but will leave the finer details of the discussions to Spurs chairman Daniel Levy.
Redknapp told Talksport: "We're interested in Joe.
"I know Daniel was talking to his people and we're interested.
"If the chairman can pull a deal off I'd love to have him at Tottenham. But it all comes down to if we can do a deal."
He added: "I don't get involved in the financial side; the chairman can do that with Joe's people. Hopefully there is a deal that can be done because we would love him at White Hart Lane."
Redknapp, who was briefly linked with the England job following the country's lamentable World Cup display, added that speculation linking Tottenham with a bid for Everton and South Africa midfielder Steven Pienaar was wide of the mark.
"I like Pienaar, don't get me wrong. But there is nothing there," said Redknapp. "I wouldn't see that as being a real goer."
The 63-year-old guided Spurs into the Champions League last season but warned a repeat in the upcoming campaign will be a tough task, with Manchester City expected to break into the top four on the back of another summer of lavish spending.
"It's going to be even harder to make the top four, that's for definite," said Redknapp.
"Manchester City could quite easily win the Premier League. If they get the players in now, they've already got the nucleus of a good team.
"They're in for [Liverpool striker, Fernando] Torres, if they end up getting him and one or two others then they can do anything. It's going to make it more difficult."
Source: PA
Source: PA