Moyes has ended up at West Ham United, following their dismissal of Slaven Bilic.
But Brazil was expecting the former Manchester United and Sunderland boss to be handed the vacant role at Rangers.
“I was expecting Moyes to go to Rangers! He’s a bluenose,” Brazil said.
Rangers are still looking for a new manager to take over, after sacking Pedro Caixinha.
The Ibrox side currently have Graeme Murty in caretaker charge, but are thought to be looking for a permanent manager from outside of the club.
Murty has done a good job since taking charge of Rangers on a temporary basis, and has breathed some optimism back around Ibrox.
At the moment Aberdeen boss Derek McInnes is the heavy favourite to take over at Rangers, with his appointment expected to go through over the next few days.
The Light Blues are currently in fourth place in the Scottish Premiership table.