The former Rangers and England defender met with Easter Road chairman Rod Petrie on Wednesday after being given permission to discuss the position vacated by Pat Fenlon last week.
Butcher, who previously managed Coventry, Sunderland and Motherwell, and turned down advances from Barnsley earlier this year, has been in charge at Caley Thistle since January 2009.
It is believed he is keen to take his backroom team of Maurice Malpas and Steve Marsella with him but Hibernian are refusing to comment on an issue which may be resolved by a compensation package.
Speaking earlier this week, 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."
In a twist of fate, Inverness travel to the capital on Saturday to face a Hibernian side under the guidance of Jimmy Nicholl, who was put in temporary charge following Fenlon's departure.