The Hammers host QPR on Sunday on the back of a narrow 2-1 loss at Manchester United last week and Allardyce is determined for his team to bounce back by getting the better of their London rivals who sit 18th with just one win so far this term.
"We're under pressure to get more points on the board," Allardyce said ahead of the Upton Park clash.
"We've played very, very well in five out of our six league games so far. The fact of the matter is seven points from those games is simply not enough.
"We have to turn performances in to results. I think we should have gained a minimum of 10 or 11 points by now. Unfortunately, we only have seven.
"At this early stage of the season you can jump places very quickly. The QPR game is almost a six-pointer. We want to go into the international break with a positive result."
West Ham find themselves sitting 13th on the Premier League table after six matches with two wins, one draw and three defeats.
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