The club currently sit bottom of the table but they are only three points adrift of safety with four games to play, and Spector is looking forward to the prospect of their upcoming games - starting with the visit to Manchester City on Sunday.
The American international, 25, told the club's official website: 'I think every player wants to play. They are massive games (coming up) and it is a big stage for us.
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'There is certainly the added pressure of that Premier League status which we definitely want to hang on to.'
Spector - whose performances in a previously unaccustomed midfield role have been a rare positive for West Ham this season - is hoping to draw on the club's experience of surviving the drop four years ago.
As the club's third-longest serving player after Mark Noble and Danny Gabbidon, Spector has picked up vital experience from being part of the side that beat Manchester United at Old Trafford on the final day of that season to secure their safety.
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'I am not sure there is a secret (to survival). It is about hard work and determination and the guts to want the ball, to want to play and to stick to your gameplan even if it is not going well for you,' he said.
'It is a very different team from the one we had that stayed up (four years ago). It is a strong team that we have right now, probably stronger than we had back then.
'But it hasn't happened for us this season but now we have to come together as a group and get some points.'
Inspirational leader and Football Writers' player of the year Scott Parker could be back to face City having missed the last two games - both of which ended in defeat.
Dog fight: Spector refuses to believe West ham are too good to go down
But Spector insisted that the club cannot just rely on Parker to keep them up, while he also warned the club of thinking that they are too good to go down,
'There could be quite a few players who could have an important role and have a massive impact on these games that we have left.
'There have been times when we have played really good football and a lot of people have been looking at us and saying 'they are too good to go down'.
'They were saying that about Newcastle a couple of seasons ago and we don't want to repeat what they did.'
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