Both sides meet in Norfolk having only won two games apiece in their opening 10 Barclays Premier League fixtures this season.
Club-record signing Andy Carroll has yet to feature for West Ham this season following a troublesome foot injury and, with only the late arrivals of Mladen Petric and a returning Carlton Cole, Allardyce has opted to play a system without a recognised striker in recent weeks.
That tactic worked to perfection at White Hart Lane as the Hammers secured a memorable 3-0 over London rivals Tottenham but, having found the back of the net only five other times so far, Allardyce wants all of his players to take some of the goalscoring burden.
"People point to the fact that we haven't got Andy Carroll or we haven't played with a front man recently but it's not just that as to why we're down on our goal count from last year," he said.
"We're down on all players scoring a few more last season than they have done this season. The main contribution to us finishing 10th last season was that most players in our squad actually scored a goal.
"Very few players in the squad didn't got a goal. Whether that was from set-plays or open play it was very, very good from the point of view that we were in the Premier League for the first season."
Despite struggling to score themselves, West Ham have been miserly in defence this campaign with only Spurs and Southampton boasting a better goals against total.
The Hammers have leaked just eight goals so far, the same number as Chelsea, and Allardyce feels such a sound base should have led to more points.
"We've done the first bit right and that's not conceding a goal," he said.
"We've had many performances as good as the one against Tottenham this year and we've had better, like against Newcastle, but we haven't won.
"We haven't got the rewards we deserved, we know that and we've got to try and improve on it. That's the only area that's concerning me and hopefully we'll put that right on Saturday.
"There's nothing else I can ask of the lads other than to try and put one or two more chances away.
"The rest of the games they've played they've been on top of the game and they've been very consistent, playing some really good football.
"We've been very resilient and it's just the final product has not delivered as much as we would have hoped and hence we've drawn a lot of games that we should have won on the basis of our performance. We will continue to strive to improve that, starting against Norwich on Saturday."
Full-back Guy Demel has played a part in the strong defensive unit developing for West Ham and the Ivory Coast international reckons just scoring once could be enough for an away win on Saturday.
"From a defensive point of view it has been really good," he told the club's official website.
"The shape, the work we do as a team and the good understanding we have as a back four are all key to us being solid.
"You just need one goal to win the game with the way we've played recently. We have to keep working as a team, deliver more balls and keep believing in our offensive quality, to make sure we force our destiny.
"Even if it's a lucky goal, it doesn't matter what it looks like as long as the ball goes in the back of the net, we keep a clean sheet and get the three points on the board."
West Ham's defence will face slight disruption following Winston Reid suffering ankle ligament damage in training earlier in the week.
The New Zealand international will not feature at Norwich with fellow defender James Collins' return from a hamstring issue coming at a good time for Allardyce.