The Reds confirmed on Wednesday morning that the 71-year-old Scot will retire at the end of the current campaign after more than 26 years in charge.
And Gill exclusively told MUTV that it won't be long before Sir Alex's successor is revealed.
"I’m not going to get into speculation on that [an exact date]... but we will move relatively quickly. I’m not going to define how quickly that is though," explained Gill, who himself leaves the club at the end of June.
"The qualities [needed] are the ones that have been inherent in Manchester United for many years. If you look at what has happened with the two most successful eras – Sir Matt Busby and Alex – they are managers who got involved in the whole aspect of the club, whether it be from the youth team up to the first team.
"All aspects of it, plus that degree of loyalty and the understanding of the football club that it’s not just what happens on the first-team pitch, is crucial to the success of Manchester United.
"Those are the sort of things we’ll be looking at. Clearly he has to have the requisite football experience, both in terms of domestic and European experience, so I think it’s a small pool but we’ll move forward."