They are not at home this weekend, but they will be favourites to overcome Championship side Blackburn Rovers in the fifth round.
A big sigh of relief greeted Rovers' 3-0 win over Fulham in midweek. It was their first in 10 league matches and helped move them eight points clear of the Championship's relegation zone.
A much-needed boost, but it is hard to see Blackburn's form really changing with five of their next seven league games against top-eight sides along with the fact that they lost star player Jordan Rhodes to Middlesbrough earlier this month just before the transfer window shut.
In the FA Cup, Blackburn have eliminated Premier League opposition at this stage in two of the last three seasons, though they went on to fall at the quarter-finals both times.
They have won the competition on six occasions - only six clubs have won it more - but the last of those was 88 years ago in 1928.
This year they have come through a pair of away ties against League Two duo Newport County and Oxford United to reach the last 16.
Recent form: WDWDLW
It is 36 years since West Ham last won a trophy, when they triumphed in the FA Cup for a third time in 1986, but some feel that the stars are aligning for them to succeed this year as they wave goodbye to the Boleyn Ground.
In the fourth round there was one of the more memorable moments in recent years when Angelo Ogbonna thundered home a 120th-minute header to prevent the tie with Liverpool going to penalties and send the Hammers through to the last 16, a stage that they have not got past in five years.
However, they should feel confident of getting past Rovers given that they have won 12 of their last 14 FA Cup games against lower-league opposition.
In the league West Ham are seventh and only one point behind Manchester United in fifth, but last weekend they had to come from two goals down to salvage a point at Norwich City.
Including that draw at Carrow Road, Slaven Bilic's side have won just one of their last 10 away games in all competitions (D5, L4).
One player who knows Blackburn pretty well from his recent days in the Championship is Michail Antonio, and the winger scored twice in his last appearance against the Lancashire outfit - a 3-3 draw with Nottingham Forest last season.
Recent form: DDWLWD
Bilic will also be without forwards Andy Carroll and Enner Valencia, who have adductor and ankle problems respectively, so Emmanuel Emenike is likely to make his first start.
Blackburn possible starting lineup:
Steele; Marshall, Duffy, Hanley, Spurr; Bennett, Lenihan, Evans, Gomez; Brown, Watt
West Ham possible starting lineup:
Randolph; Oxford, Collins, Ogbonna, Cresswell; Noble, Song, Obiang; Moses, Emenike, Payet
Head To Head
West Ham are unbeaten in their last 11 matches against Blackburn in all competitions, although the last five meetings have all been draws.
On the four occasions that Blackburn have hosted West Ham in the FA Cup they have won just once.
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We say: Blackburn 0-2 West Ham
Having knocked Liverpool out in such memorable fashion, West Ham will not want to go out with a whimper here and Blackburn should not pose much of a threat for Bilic's boys.