The Hammers visit White Hart Lane sitting in seventh place in the Premier League and two points above their hosts, but Allardyce admits his side's best may still not be enough.
After the trip to north London, Allardyce's men then face Chelsea, West Brom, Everton and Arsenal, all of whom are in the top six after 12 games.
He said: "We know we are in a very good position but this run of fixtures could cause us a few problems.
"I think we will play well and I do not think we will let ourselves down but we could play our best in the next four games and lose all four. If Tottenham, Manchester United, Chelsea and Liverpool are all at their best then it is going to be very difficult for us.
"I think that the players at this football club worked very hard last year to play in fixtures like the ones coming up.
"It is always a massive challenge for a newly-promoted side to face the likes of Manchester United but I think it is a challenge that the players look forward to.
"It is all about accumulating the points and continuing to pick them up as often as possible. If we carry on to do this then we will move towards a points total that will ensure safety."
Allardyce is hoping the London derby ends with their current narrow advantage over Spurs widening by a further three points.
"I think that local derbies are always very special," he said.
"They are games that the fans and the players always enjoy and our hope is that we can come away with a few positive results. I hope we can take our game to Tottenham and cause them a few problems."
According to Allardyce his squad is brimming with confidence ahead of the trip to White Hart Lane and believes it is his task to keep the momentum going.
When asked if a win over Spurs would give West Ham confidence for the remainder of the season Allardyce said: "I think we have got it now.
"We are a third of the way through the season and we have produced a high level of performance so we have the confidence and belief to keep that going.
"I have got to keep driving the players on and I have to strive to make the players even better. My aim is to ensure that the players continue to play well and improve.
"The squad was put together in the summer and a lot of new players came in so we will continue to grow as we get to know each other better."