The Hammers have only spent three weeks out of the drop zone this season, after wins over Fulham and, ironically, Sunderland in Howard Wilkinson's first game in charge.
To add to the irony, it was West Ham's closest rivals Bolton who beat Sunderland last week to add to the pressure on West Ham, but that is something that Paul feels the Black Cats won't be under playing at Upton Park.
"Obviously they suffered a bad defeat for them last weekend at home," he says "but I think they might be in a situation away from home where there is a little less pressure on them, so they will come with a free will about them to really have a go and try to win the game.
"They are still in the shake-up and with eight games left there are a lot of points there, so I am sure they still believe they can do it.
"But time is running out for all of us so each game becomes that bit more vital."
Sunderland's first objective is to get off the bottom, which they would do if they won and West Brom lost at Birmingham - a result which would hardly favour the Hammers either.
But the Mackems also have other considerations. Defeat for them would mean eight successive Premiership defeats - just one short of the club's league record of nine consecutive defeats suffered between November 1976 and 1977.
If they let in three goals, as they did in the corresponding fixture last season, they will then have conceded 250 from five Premiership seasons - though that is hardly likely to form the basis of Mick McCarthy's team talk.
But getting off the bottom would present the same psychological boost for the visitors as getting out of the drop zone would for West Ham - and only a win would achieve this on Saturday.
"The aim is to be out of the bottom three as soon as possible and if it can be done this weekend it would be fantastic," says Paul.
"As long as it is done before the end of the season we will settle for that, but it is an opportunity again and we have to approach it in the right manner."
Paul has noticed a change in mood at Chadwell Heath recently, and he adds: "You can't help but notice that winning football games breeds confidence and belief in players. Without doubt we have noticed it and there has been a far greater resolve around the place - and long may that continue.
"We need every West Ham United player for whatever role is put on them in the last eight games."
Looking back at the Everton match, he says: "It takes two sides to make a really entertaining game and Everton played a correct game, a very strong one which they have been doing all season.
"It was a tough Premiership game and our lads have to take a lot of credit for knuckling down, and withstanding.
"There were few chances either way and Trevor's chance was possibly the best of the game, but Richard Wright made an excellent save and the chance was gone.
"It was a very hard-earned point and we were pleased with it."
West Ham even survived a triple substitution in the second half, and Paul adds: "This is where you have done your homework and we were prepared for what possibly could happen, though we didn't expect it to happen 10 minutes into the second half.
"We kept them at bay, not many teams will do that, as they are having an excellent season. David Moyes is doing a super job with Alan Irvine and they have really revitalised.
"It is a fantastic achievement to get to fourth and great credit must go to them; their home record has been super and to get a point was very good for us."
Paul is full of praise for the defence's performance last Saturday, and hopes they can keep another clean sheet against a side who, with 19 goals scored in 30 league games, have the lowest scoring record of any side in the five English and four Scottish leagues.
"I thought our two centre halves were excellent against Everton, in fact the whole back four was - and David," says Paul.
"Now we have a home game and we have got to go into it full of confidence; we have to approach it like we have approached the last three or four games.
"We have got to be very positive and work very hard to shut the opponents down."