The Tykes, who prop up the Championship table, are on the lookout for a new boss having sacked David Flitcroft at the end of November, with assistant Mellon currently in caretaker charge.
Flitcroft himself was promoted from caretaker-boss less than a year ago to replace Keith Hill at Oakwell, an appointment which came at the end of a very public process which saw both Sean O'Driscoll and Terry Butcher turn the job down.
But chief executive Ben Mansford insists Barnsley will complete their ongoing search behind closed doors this time around until an appointment is made.
"Last year there was a very public process that occurred around who was and wasn't going to be the next Barnsley manager," Mansford said in a statement on the club's official website.
"That bred at times, picking up from David and Micky and Martin (Scott, former technical coach), a bit of uncertainty around the place.
"We want to do our business in private, in football terms it is the best way to do it. We are getting on with what we feel is right for the football club.
"As and when there is clarity as to where we are going and what's going to happen, we'll let the fans know at the earliest opportunity."
Flitcroft has this week made an immediate return to management at League Two club Bury, with Mansford admitting he hopes the former Shakers midfielder enjoys every success at Gigg Lane.
Mellon claimed an impressive 2-1 victory at Brighton in his first match in temporary charge of Barnsley before falling to a narrow defeat at leaders Burnley last weekend.
And Mansford added: "Our plan was always to allow Micky to be his own man.
"The changes to the staff have created an opportunity for him to do that.
"And I'm absolutely delighted that Micky prepared the team sufficiently well enough with real expertise and experience to make sure we got a very, very good away win at Brighton.
"We always wanted Micky to have the opportunity to be his own person. He's earned the right to do that and that's where we currently sit at the moment."
Meanwhile, Mansford has revealed Barnsley's search for a new club crest has taken a back seat behind on-field matters and the managerial search.
But after fans were invited to cast their vote, he revealed it is a close call between one unnamed new design and keeping the existing badge.
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