The Hammers have two wins all season - on the opening day against Cardiff and, memorably, at Tottenham - and lost 3-0 at home to Chelsea on Saturday night to sit in a perilous position.
Allardyce saw his side jeered off at the interval and full-time of the London derby and insists the supporters are entitled to vent their frustrations following a fourth home loss of the season, the same number his side had in the whole of last term.
"If they don't see the sort of determination and passion and desire they think they should get to show their disapproval," Allardyce said.
"I thought the players tried as hard as they can, but they're playing with a bit of fear at the minute. And that makes them look like they're not playing with the passion that is normally seen."
Two further London derbies follow for the Hammers, against Fulham and Crystal Palace, with an upturn in fortunes crucial heading into the busy festive period.
Allardyce insists his side must block out the consequences of their performances.
"When people say there's a pressure game at the top - Manchester United versus Arsenal or Chelsea versus Manchester United or Manchester City - it's not as big a pressure as when you're in the bottom three or four," Allardyce added.
"It's much greater pressure because it's all about 'relegation this', 'relegation threatened that', all the money you lose.
"That becomes a constant flee in the players' ear. They've got to dismiss all that, keep a clear mind, keep a clear head and focus on playing how they know they can play.
"It's my job to pick the players up and remind them about previous performances we've had home and away this year, make sure we get to that level and stop those fears creeping in."
Yet, following the failure to build on the 3-0 win at Spurs, Allardyce has also reminded his side of their predicament.
He added: "I said to the players because you've thrown away the wins you should've got, you've put yourselves under so much pressure now that you're going to find it really tough to play football as easy as it was for you a few weeks ago. It's now becoming tougher and tougher.
"We've got to try to dismiss all of those negatives and get back to a proper positive attitude this week."
Allardyce again bemoaned the continued absence of Andy Carroll as the England striker continues his rehabilitation from a heel injury which has yet to see him strike a ball in anger this season.
The Hammers boss was asked about strengthening in the winter transfer window, but he is eager to get there first.
"Andy was due back three weeks into the season, not still out now," said Allardyce, who again reported the Hammers missed out on four strikers before the close of the summer transfer window.
"It's been a double whammy for us that we've had to live with.
"It's alright saying going into the market in January, (but) there's too many games between now and January we've got to make sure we get points from."
Joe Cole was hauled off before half-time against his former club and reacted with frustration.
Allardyce attributed Cole's underwhelming displays to niggling injuries.
"Getting back to top form is playing on a consistent basis and that's been a struggle for Joe," Allardyce said.
"We've had him at top form, but not for long enough, which is a shame for the talent he's got."