The former Bolton, Newcastle and England manager cited United's side in 1974 that, against all expectations, went down to the second tier.
With Solskjaer overseeing the club's worst start in 30 years, Allardyce has used the side from 45 years ago as a warning to the current crop that they are not too good to be relegated.
He told talkSPORT: "It isn't beyond the realms of possibility that they could catastrophically fall into more problems if confidence goes and injuries stay like they are.
"Players tend not to get as fit as quick as they would do when things are going well, that's an absolute fact. There's the disturbance of [Paul] Pogba wanting away and who will pop up next saying, 'I see this going the wrong way, I want to leave?'
"Manchester United do not have enough goals and you can see it becoming a struggle. Hopefully it's not a struggle down the bottom end, but they're certainly heading that way."