Robbie Keane - who has been playing with Cole at LA Galaxy - is expected to begin talks with Birmingham this week over a move to the club, Sky Sports News HQ understands, which could persuade Cole to follow him to St Andrew's.
Reports had linked the 107-capped former England global with a move to the Championship club and Redknapp feels that despite Cole's advanced age he would still be a major asset to a club chasing Premier League promotion.
Cole, 36, has played for Roma and the Galaxy since leaving Chelsea in 2014, while Keane, 37, has been a free agent since leaving the Californian side previous year.
With Redknapp set to take Brum fans for a trip down memory lane with these move for Keane, Cole and Downing, it'll be worthwhile seeing whether the three still have it in them to perform against some of the country's biggest clubs.
"And for me he was possibly the best left back in the world a few years ago at one time when he was at Chelsea. Wage-wise I don't know where we would go with it, I haven't got a clue what he earns, and he belongs to Middlesbrough, but I think they've said he can go on a free, so he's of interest". I don't know what Ashley's plans are.
"Robbie Keane is a proper player; a great professional and infectious character", said Redknapp. When you have a chance to get someone like him, if he's available, he has to be of interest. He's a great trainer.
Whether we could afford him I don't know. "I'm sure he'd fight tooth and nail to get a regular place in the team".
Having just ended their pre-season tour to Austria with a morale-boosting win, Birmingham must do clever business over this transfer window if they want to challenge the top six spot in the forthcoming season.