The Englishman took a step closer to being confirmed as Pat Fenlon's replacement after Hibs and his club Inverness agreed a compensation deal.
Butcher and his number two Maurice Malpas - who has turned down the Caley Thistle job to join Butcher at Easter Road - are in the process of thrashing out personal terms.
A statement from Hibernian read: "Hibernian FC confirmed (on Monday) that compensation has been agreed with Inverness Caledonian Thistle FC.
"This means Manager Terry Butcher and Assistant Manager Maurice Malpas now have permission to speak to Hibernian FC to agree personal terms."
Chairman Rod Petrie added: "We are grateful to Inverness Caledonian Thistle for the professional and courteous way in which the discussions between the clubs have been conducted.
"We now look forward to welcoming the new management team to Easter Road Stadium."
News of Butcher's likely departure was met with disappointment by Inverness, who also confirmed Malpas rejected the opportunity to take over as manager.
Chairman Kenny Cameron said: "It is a major disappointment to the club and its supporters that they have taken this decision.
"I'd like to put on record our appreciation of the excellent job they have done at Inverness since Caley Thistle provided them with the opportunity to join the club in January 2009.
"Maurice was offered the manager's role but declined.
"We would, however, like to record our sincere appreciation to him for taking the team on Saturday in what was a resounding victory."
Meanwhile, Hibernian youngster Danny Handling backed Butcher as the man to transform the fortunes of Hibs.
The midfielder said: "I would be happy if he took over because he's a good manager.
"You see what he has done at Inverness, he's got them high up the table and they are in a (Scottish League Cup) semi-final as well.
"He has proved he is a good manager at that club.
"Our team is low on confidence. I think we are the lowest-scoring team in Scotland and that speaks for itself.
"But whoever comes in could change that within one game and get us firing in goals and winning games and back to our usual selves."
Butcher looked on from the Easter Road stand on Saturday as his Inverness side claimed a 2-0 win over his potential new club to move into second spot in the Scottish Premiership.
Handling added: "They came to Easter Road on Saturday, which is a hard place to come to for many teams, and, to me, they bossed the game and deserved to win.
"That shows how good a job he has done up there and, if he does get the job, what he could potentially do."
- Sports & Recreation
- Terry Butcher
- Maurice Malpas
- Caley Thistle
- Hibernian FC