After making an impressive start to their first season back in the Premier League, The Hammers have stuttered of late, having only taken one point from their last three games, suffering successive home defeats in the process.
Allardyce knows his side cannot rest on their laurels in an unforgiving Premier League, and is fully aware they could easily find themselves battling at the wrong end of the table.
"We're still not away from the danger zone, we're actually slipping towards it," he told the Daily Star.
"You drop one more place, then one more place more and that pressure grows week by week. And believe me, you don't want to feel that pressure.
"We've got a few lads who have felt that pressure last time we got relegated - Mark Noble, Jack Collison, James Tomkins, Carlton Cole - and they don't want to feel it again."
One shining light of late has been the form of Cole in the injured Andy Carroll's absence, and with his recent dismissal against Everton overturned, the Hammers boss challenged the striker to carry on where he left off.
"We're in our most difficult period in terms of results and injuries. But the best news is Cole's red card being overturned," he added.
"We are very pleased to have him with us for a very important time of year. He's been far more consistent in my time here than I expected, and when he's on song I enjoy watching his ability.
"The question is whether he's done it consistently enough with the ability he has."