The Irons moved on to 37 points following last weekend's battling goalless draw at Liverpool and to within touching distance of the benchmark which usually sees teams avoid the drop.
While West Ham may have gone down in 2003 despite amassing some 42 points, Allardyce believes given the battles below them in the table, just one more win should be enough to retain top-flight status.
"I know mathematically people will say we can still be caught and somebody could still overtake us, but I find it difficult to believe that anybody would," Allardyce said.
"Over the 20-odd years (of the Premier League) there are only a few cases of teams going down with 40 (or more), but it has been a long time since that was actually the case."
Midfielder Gary O'Neil feels the players have what it takes to get the required result tomorrow and then to push on up the table.
Speaking to West Ham TV, O'Neill said: "We are looking for one more win and hopefully it can come this week.
"I don't think we're counting ourselves as safe yet because we're only six points above the bottom three.
"It is not completely done and dusted but we're looking to hit as many points as we can anyway - we are not going to get to 40 and stop.
"We are going to push on from there and try to get into the high 40s if we can
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