Aaron Doran and Graeme Shinnie fired the hosts into a two-goal lead before
Darren Barr halved the deficit 14 minutes from time.
Regarding the vacant seat at Easter Road, from where Pat Fenlon departed on Friday, Butcher said: "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."
Following the victory at the Caledonian Stadium, Butcher added: "In the end it was a great win for us because we were put under a bit of pressure in the second half.
"Kilmarnock switched their system round and made it difficult for us. However, the boys saw it out very comfortably."
However, Kilmarnock boss Allan Johnston says his team performed well after the break.
He stressed: "With the conditions in the first half it was difficult, but Inverness were probably the better team.
"However, I felt that we were by far the better team in the second half.
"We put them under a lot of pressure and were unlucky not to get an equaliser."