Pat Fenlon resigned from his post at the Edinburgh outfit on Friday, after a Scottish League Cup defeat to city rivals Hearts.
Hibs currently sit seventh in the Scottish Premiership, seven points and five places behind Inverness, who are second to champions Celtic.
Butcher led his side to fourth last season, narrowly missing out on a place in the UEFA Europa League, while the Highland club also reached top spot in Scotland's top flight for the first time in their history this term, remaining at the summit for the first eight rounds of the campaign.
Butcher turned down a chance to manage English Championship side Barnsley last season, and Inverness have blasted rumours that the former England captain could be on the move.
A club statement read: "Inverness Caledonian Thistle FC made it clear this evening (Monday) that, despite speculation to the contrary, no club has contacted them to ask for permission to approach manager Terry Butcher.
"Terry took training this afternoon and following the session he assured the club chairman that he has not spoken to any other club."
Butcher has been in charge at Caledonian Stadium since January 2009 and the club expect him to remain in situ for the foreseeable future.
Chairman Kenny Cameron added: "We are extremely unhappy about a story in relation to our manager. Wherever this has come from, it is completely untrue.
"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."
