The Scot has been widely discussed as Ronny Deila's potential successor at Celtic.
As shown by Sky Bet, Moyes is the bookies' favourite to become the Bhoys' next boss.
And Irvine is quoted as saying by the Record: "Things are changing in football in terms of the structures of clubs. The manager used to manage.
"He was in charge of the groundsman, the cooks, the cleaners plus the players. Clubs are so big now it’s hard to control all of that.
"But I can safely say David would not welcome anyone interfering with the selection of players, either to play, or in recruitment. Recruitment selection would have to be his, I’m quite sure."
It's a crossroads for Celtic when it comes to their next manager.
Boards in football are often stubborn and continue down the same route regardless of what their fans say. It's what Randy Lerner did at Aston Villa and look at them now.
If Celtic want to continue making a profit and selling players on for money, the board can do just that. But they can't then complain if Rangers overtake them once again - nor can they pin the blame on any future managers when things go wrong.
Their alternative is to start investing in transfers more and handing control of signings to someone like Moyes or whoever it is that ultimately takes over at Parkhead.