The Foxes have accepted a £50,000 offer for the 21-year-old, and City expect to agree personal terms with the player in time for him to play at Torquay on Saturday - Taylor's first game in charge.
The new City manager said: "As soon as I discussed taking the Hull job, I told the chairman that Damien was a player I wanted to sign.
"I'm delighted he has backed me and hopefully everything will be sorted out today. We have agreed a fee with Leicester and it's just a matter of sorting out the personal side of things with the player. I don't think there will be any problems and I'm very confident he will sign."
Taylor originally signed Delaney from Cork City two years ago when he was at Filbert Street, but he has only started a handful of games. He has been loaned out to Stockport and Huddersfield, and his move to Boothferry Park cuts short a spell at Second Division Mansfield, for whom he has played 7 games this season.
He comes highly rated from Taylor, who was clearly keen to sign him again: "I brought Damien across from Ireland when I was at Leicester. He's a big lad but he can play. He's very strong, has a good left foot and I'm sure he'll do a good job for us."
Meanwhile, Taylor adds that he would have moved for Delaney whether or not City had beaten Rochdale on Saturday. After seeing the Tigers for the first time he said: "It was a very encouraging performance and has given me a lot to build on."
Taylor also compliments Steve Burton on his impressive first team debut, though he may lose his place to Delaney at left-back. Shaun Smith and Mike Edwards will still be missing.