West Ham manager Sam Allardyce has challenged his team to all but secure Barclays Premier League safety with victory at Southampton on Saturday.
The Hammers moved on to 37 points following last weekend's battling goalless draw at Liverpool and to within touching distance of the benchmark which usually sees teams avoid the drop.
While West Ham may have gone down in 2003 despite amassing some 42 points, Allardyce believes given the battles below them in the table, just one more win should be enough to retain top-flight status.
"I know mathematically people will say we can still be caught and somebody could still overtake us, but I find it difficult to believe that anybody would," Allardyce said.
"Over the 20-odd years (of the Premier League) there are only a few cases of teams going down with 40 (or more), but it has been a long time since that was actually the case."
Midfielder Gary O'Neil feels the players have what it takes to get the required result tomorrow and then to push on up the table.
Speaking to West Ham TV, O'Neill said: "We are looking for one more win and hopefully it can come this week.
"I don't think we're counting ourselves as safe yet because we're only six points above the bottom three.
"It is not completely done and dusted but we're looking to hit as many points as we can anyway - we are not going to get to 40 and stop.
"We are going to push on from there and try to get into the high 40s if we can."
Former Portsmouth player O'Neill added: "Southampton are in great form and they have had some fantastic results at home - I think they have beaten Manchester City, Chelsea and Liverpool at home recently so it will be a tough game because they've got some good players.
"However, our away form has picked up with a win at Stoke and draw at Anfield after five straight defeats on the road. Hopefully our improved form can continue."
At one point it looked like Southampton would struggle to stay clear of the relegation battle. However, successive home wins over Liverpool and Chelsea followed by a win at Reading have moved Mauricio Pochettino's men swiftly away from danger and level on points with the Hammers, but a place higher because of goal difference.
It is a run of positive results from attacking football which has impressed Allardyce.
He said: "At home, we all want to use a pressing game because it creates a much more active and quicker game that fans want to see. They don't want to see us playing with a slow tempo and a slow build up in the Premier League.
"Certainly when you're the home team the emphasis is on you, so actually pressing high up the pitch is a good way to get the players on a high tempo and then of course get the fans cheering for the team."
Allardyce has to decide whether to recall Andy Carroll after the on-loan forward was not available for selection against his parent club last weekend.
The Hammers boss, however, knows Southampton present plenty of attacking problems themselves.
"(Rickie) Lambert and (Jay) Rodriguez without a doubt, and (Adam) Lallana if he plays, all three of them have mastered the challenge of the Premier League brilliantly," he said.
"Lambert has been doing it right from the very start. He has scored goals in every division he has played in now, which is a magnificent achievement.
"Rodriguez, of course, having had time to settle from Burnley, has shown the sort of goalscoring form that he managed before.
"Lallana has always been a good player in my eyes so that's a very dangerous front three, so we will need to be as good defensively as were against Liverpool."