Villa manager Paul Lambert has come under increasing pressure after a run of six straight top-flight defeats that has left his side just a point above the relegation zone.
There were plenty of positives for Lambert in last weekend's 2-1 loss to Tottenham, though, not least the fact Andreas Weimann's opener ended a barren run in front of goal that lasted 547 minutes.
However, the game swung on Christian Benteke's dismissal for raising his hand into Ryan Mason's face 25 minutes from time, and Spurs duly took advantage with late goals from Nacer Chadli and Harry Kane.
And Allardyce has called on his in-form side to extend Villa's woe.
"We have to make sure we try and continue their poor run," he said. "He'll [Lambert] remember when they pulled off a result at Liverpool [Villa's last Premier League victory on September 13] and say that's what they're capable of, so I'd imagine it'll be a very difficult game."
Allardyce faced uncertainty over his own future during a difficult 2013-14 campaign and admits he has sympathy for Lambert's situation.
"Always [you sympathise], because there's a huge pressure that comes upon you when you lose so many games on the trot," he added.
"He must be wondering what he's doing wrong at the moment, because I watched the game against Tottenham last Sunday and thought how unlucky he was that he lost that game."
In contrast to Villa, West Ham are in fine form in the Premier League.
Three straight wins were followed by a battling 2-2 draw at Stoke City last weekend, and they sit fifth in the table.
Stewart Downing scored the equaliser in that match and the midfielder's excellent recent form has been rewarded with an England recall.
In a further boost for West Ham, striker Andy Carroll could return to their matchday squad.
The 25-year-old has not featured since May due to a heel injury, but he came through a behind-closed-doors fixture with Southend United unscathed on Wednesday.
Carroll will have to work hard to regain his place in the team, though.
Diafra Sakho has six goals in seven Premier League appearances for West Ham and Allardyce is hopeful he can play after missing the Stoke game with a shoulder injury, while defender Winston Reid (leg) could also come back into contention.
Villa too have received a fitness boost with the news Philippe Senderos could return from a three-match absence due to a thigh injury, but the game comes too soon for fellow defender Alan Hutton (ankle).
Lambert must also rejig his striking options with Benteke set to start a three-match suspension.
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