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.