Gill, who was speaking to the club's official television channel MUTV, also insisted that the successor will need "domestic and European requirements".
"We have to look forward now and I think any manager coming in will inherit a great squad and great staff around him. It is a dream job. I won't speculate on the announcement but I think we will move relatively quickly," said Gill.
"In Sir Matt Busby and Sir Alex we got in managers who got involved in all aspects of the club. That degree of loyalty is crucial to the success of Manchester United. He (the next manager) needs the required football experience in terms of domestic and European requirements, so it is a small pool."
Everton boss David Moyes has emerged as the early favourite.