Billy Stark believes the Scottish champions' squad has gotten too big.
A total of 23 players arrived during the Norwegian's time in charge and Stark believes that having too many bodies can have a detrimental effect on dressing room morale.
"I think Celtic need an established manager,” he said, as quoted by the Scotsman. “The size of the club means they need an experienced man who knows how to handle a dressing room.
"Ronny has said what he’s said about how it’s run but, for me, with 30-odd players in that dressing room, I’d be waking up in the middle of the night worrying about how I’d control that number.
"It’s impossible to have over 30 players and keep everyone happy. An experienced boss wouldn’t deal with that number because, these days, that’s what management is about.
"I would guess the first thing they’ll do is cut the numbers back."
Deila was still in his thirties when he arrived at Parkhead two years ago and the current list of bookies' favourites suggests Celtic will heed the advice of Stark, who spent six years with the Bhoys as both a player and as Tommy Burns' number 2.
46-year-old Paul Lambert currently leads the betting, with 53-year-old David Moyes and Brendan Rodgers, who has eight years' managerial experience though a relative youngster at 43, also highly placed.
Celtic, who have just Charlie Mulgrew and the on-loan Tyler Blackett out of contract this summer, have agreed a deal for Norweigian midfielder Kristoffer Ajer already, while Nadir Ciftci, Anthony Stokes and Stefan Scepovic are among the many names returning to the club from temporary spells elsewhere.