The current Leicester City manager won eight out of eight trophies before jumping at the chance to join the Foxes in late February of this year.
Allardyce believes similar success can be achieved by any boss going to either half of the Old Firm as he expects Steven Gerrard to soon do the same at Rangers.
He told talkSPORT: "We can all go to Celtic and win trophies.
"It will be Rangers soon the way they are going under Steven Gerrard .
"Fans expect you to win trophies and if you don't do that then you'd get the sack."