The Hammers boss says that there are historical examples of sides in their position pulling off a great escape, and he thinks it could happen again this term.
"I wouldn’t like to say which clubs will go down but I predict at least one of the bottom three will come out of the pack," Allardyce said.
"Alan Curbishley did it here, Roy Hodgson managed something similar with Fulham, likewise Roberto Martinez at Wigan and I did it with Bolton.
"We did it last season but earlier when we won four on the trot which took us out of the danger zone."
If Allardyce is correct in his prediction, then Hull City and Sunderland are the sides which look to be in the most danger of getting caught.
Hull and Sunderland are both in poor form at the moment and only hold a slender advantage over the current bottom three.
Hull are outside the drop zone on goal difference, while Sunderland are just one point clear and are well within range for Leicester, QPR or Burnley to catch.
Allardyce's West Ham side, meanwhile, are currently well clear of relegation struggles and he will be glad to be watching on as a neutral, rather than being in the thick of the survival fight.