The England winger has been told by Garry Monk that he can leave the Riverside Stadium this summer, and the Blues manager registered his interest in the 33-year-old a month ago.
On Friday, however, Redknapp told talkSPORT: "It's still dragging on a bit.
"We are interested but we just can't manage to come to an agreement at the moment with Middlesbrough. They have been fair, but we can't get it over the line.
"But I would like to bring him in. He has got quality and he would certainly help us an awful lot."
Downing has two years left on his Boro contract.