The Inverness boss has been in talks with Hibs chiefs over replacing Pat Fenlon at the helm following his departure on November 1.
Butcher could be unveiled in the next 48 hours if both clubs can reach agreement on a compensation deal.
And, if the Englishman does take over the hotseat, Handling has backed him to repeat the success he has enjoyed with Caley Thistle.
The midfielder said: "I would be happy if he took over because he's a good manager.
"You see what he has done at Inverness, he's got them high up the table and they are in a (Scottish League Cup) semi-final as well.
"He has proved he is a good manager at that club.
"Our team is low on confidence. I think we are the lowest-scoring team in Scotland and that speaks for itself.
"But whoever comes in could change that within one game and get us firing in goals and winning games and back to our usual selves."
Butcher looked on from the Easter Road stand on Saturday as his Inverness side claimed a 2-0 win over his potential new club to move into second spot in the Scottish Premiership.
Handling added: "They came to Easter Road on Saturday, which is a hard place to come to for many teams, and, to me, they bossed the game and deserved to win.
"That shows how good a job he has done up there and, if he does get the job, what he could potentially do."
The 19-year-old admits he would like to see a new manager in place as soon as possible and end the uncertainty surrounding the Leith outfit.
He said: "We don't know what's happening. There are a lot of players who are confused and lacking confidence.
"A new manager could change that completely. We are just biding our time to see who gets the job.
"Hopefully he takes a shine to the boys and he can build a good team and start climbing that table.
"The boys are just trying to go out and play their usual game every weekend. There's not really a structure to the team because there is no manager.
"Everything is in place for the team to do well - we just need that final piece of the jigsaw to complete us."
Handling has been called up to the Scotland Under-21 squad and could make his debut at that level when they host Georgia in a European Championship qualifier on Thursday.
And the teenager says he is grateful for the break from domestic football while the club resolves its managerial situation.
He said: "There is a lot of pressure at the club just now.
"The fans want a lot more and the players want a lot more from themselves as well.
"We don't have a manager and we don't know who's coming in so we have to see how that's going to go.
"To get away from it is a good opportunity for me to prove myself here on a good stage. Hopefully the new gaffer likes what he sees."