The Englishman has been tipped to replace Pat Fenlon at the Easter Road helm, following his departure on November 1.
Butcher, along with his number two Maurice Malpas, has been given the go-ahead to thrash out personal terms with Hibs chiefs.
A statement from the club read: "Hibernian FC today confirmed 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."
Hibs are hoping to unveil Butcher as their new manager at a press call on Tuesday afternoon.
Chairman Rod Petrie said: "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.
A club statement read: "Inverness Caledonian Thistle FC regret to announce that, despite every effort to persuade them to stay, Terry Butcher and Maurice Malpas have indicated their wish to relinquish their posts as manager and assistant manager of Inverness Caledonian Thistle.
"Compensation has been agreed with Hibernian FC."
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."
- Sports & Recreation
- Maurice Malpas
- Terry Butcher
- Inverness Caledonian Thistle
- Hibernian FC