Terry Butcher is staying in Inverness after turning down the chance to manage Barnsley.
Caley Thistle gave their manager permission to discuss a move to the Championship club, who parted company with Hill earlier this month.
Butcher travelled south for talks but could not be persuaded to walk away from Inverness in the middle of a promising season.
He told the club website: "I was pleased that Caley Thistle gave me the opportunity to talk to Barnsley, who were excellent to deal with.
"But I have decided that my future lies at Inverness. We have been going extremely well to be second in the SPL, still in the Scottish Cup and with a League Cup semi final to look forward to later this month.
"I have also brought people to this club and, on reflection, feel that I should remain with them to try and continue the good work. My focus is now firmly on that."
Chairman Kenny Cameron said: "We're delighted that Terry Butcher is to remain with us and the fans will greatly welcome the news.
"When Barnsley made their approach we felt that out of courtesy we should allow Terry the opportunity to meet with them. But it is a huge relief that he has opted to stay."