This comes amid growing expectation that David Moyes will be confirmed as the new man in charge by the end of this week.
After confirming Ferguson's exit this morning, it is thought United wish to announce the 71-year-old Scot's replacement by the end of the week.
Although Real Madrid coach Jose Mourinho has been tipped to take over in some quarters, it is thought Everton boss Moyes is in line to land what Gill describes as a "dream job".
"The qualities are the ones that have been inherent at Manchester United for many years," Gill told MUTV.
"Our two most successful eras were with managers who got involved with all aspects of the club, from the youth team to the first team, to get that degree of loyalty and understanding of the football club.
"Clearly he has to have the requisite football experience, both in terms of domestic and European experience.
"It is a small pool."
Moyes' European experience is not extensive.
He has only guided Everton through four European campaigns and never went beyond the last 16 phase in any competition.
However, Ferguson has never had any doubts his fellow Scot was capable of managing at the very highest level.
And Gill confirmed Ferguson, and Sir Bobby Charlton - who has previously suggested Mourinho was not the type of character would sit easily at Old Trafford - would be asked for their input.
"No-one knows what managing Manchester United is about more than him," Gill said.
"The board will take his counsel and that of Sir Bobby Charlton in order to get the right person to take it forward."
Charlton would be acutely aware of the problems it caused - first for Wilf McGuiness, then Frank O'Farrell - when Sir Matt Busby remained active at Old Trafford following his retirement as manager.
The spectre of Ferguson is going to hang over Old Trafford for a long time.
But Gill is confident the Scot will not be a negative presence.
"Sir Alex will be a great asset to the club," Gill said.
"He will know when he should and should not be involving himself.
"Everyone can rest assured the new manager can get the space and opportunity to do his job without interference.
"Alex will not make that mistake. Of that we can be certain."
As Gill is also departing in the summer, it represents a significant test for new chief executive Ed Woodward.
"Ed Woodward has the opportunity to develop a relationship with the new manager," Gill said.
"I am sure he is capable of doing that for the betterment of Manchester United.
"Alex and I will still be there but on the football club board.
"It is a new era but I am confident the structure we have in place gives a great foundation for that relationship to happen."