Cole, a free agent after leaving Chelsea, is understood to have received lucrative offers from Manchester City and Arsenal, but Spurs are strong contenders for his signature.
Redknapp has already held discussions with Cole and, having offered him the chance to play Champions League football in his favoured central position, he is looking to seal a reunion with the player he signed for West Ham as a youngster.
In an exclusive interview with Soccernet, Redknapp said on Monday: "I managed Joe before, and I signed him when he was 11. We have always had a good relationship. I know what makes Joe tick and it's his love of the game, and his need to be playing it. That is what I am offering him.
"I have spoken with Joe at length and I would like him and, when we talked, he appreciated my view that Tottenham is now a club going places.
"I would like Joe here with me, but it is now up to the club and up to Joe and his agent and his dad, and our chairman, Daniel Levy, is talking to them to try to do a deal. If it is beyond us in financial terms I can appreciate that but I am hoping other factors will come into play to make up Joe's mind."
Cole, 28, is believed to be keen to remain in London, but no decision has yet been made, with Cole's advisors continuing to explore the situation.
Meanwhile, Tottenham have confirmed that midfielder David Bentley will miss the pre-season tour of the USA.
A statement on the club's official website read: "MRI scans on Monday have confirmed that David Bentley has damaged right ankle ligaments. The midfielder was forced off with the injury in the first half of our pre-season friendly against Bournemouth on Saturday and as a result will now miss our tour of the United States to receive further treatment at Spurs Lodge."