The move is likely to see Butcher take charge at Easter Road as according to BBC Scotland he has already agreed terms at the Edinburgh.
Inverness chairman Kenny Cameron said: "Terry has been informed of the situation and has expressed an interest in meeting with Hibs officials. The club has, with great reluctance, granted him permission to do so. We will, however, continue to do everything possible to ensure that Terry remains our manager."
Butcher was linked with the Hibs job soon after Pat Fenlon resigned on Friday afternoon following less than two years in the post.
The former England and Rangers captain turned down a move to Barnsley last season before leading Caley Thistle to a fourth-place finish in the Scottish Premier League.
Last week Butcher led the Highland side to a second League Cup semi-final against Hearts in two seasons and saw his side jump to second place in the Scottish Premiership with a 2-1 win over Kilmarnock.
Cameron, whose club face Hibs at Easter Road on Saturday, added: "Given our current position in the SPFL and the fact that once again we have a cup semi-final to look forward to, it is inevitable that if any managerial positions arise, our manager will be on the wanted list.
"We have a game on Saturday and it's important that our club's focus is on taking three points to maintain our challenge at the top of the league.
"As with the Barnsley approach in January, we will be making no further comment until we're fully aware of the outcome of discussions."
- Sports & Recreation
- Kenny Cameron