Caley Thistle chairman Kenny Cameron declared himself "extremely unhappy" with the reports and said they were "completely untrue".
A club statement stated that Butcher had taken training on Monday afternoon and assured Cameron afterwards that he had not spoken to any other club.
The statement read: "Inverness Caledonian Thistle FC made it clear that, despite speculation to the contrary, no club has contacted them to ask for permission to approach manager Terry Butcher."
Cameron added: "There has been no approach to this club for Terry Butcher's services and we look forward to him continuing the excellent job he has been doing in Inverness."
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.
After Saturday's game, Butcher said of reported interest from Hibs: "All I would say is that it's just speculation as far as I'm concerned.
"We don't know how good this club could be, where it could go, or what's at the end for us.
"At this moment in time, I'm staying here - that's it."