The London club lost 3-1 at Newcastle on Sunday to stay in the relegation zone and two points from safety with one match left, against Leeds at their Olympic Stadium home on the final day next Sunday.
The Hammers will be relegated before then if Tottenham beat Chelsea away on Tuesday in the first of their two remaining games.
A draw at Stamford Bridge would also likely be enough for Spurs, given a far superior goal difference, before their final match at home to Everton.
Newcastle scored twice in the first 19 minutes, through Nick Woltemade and William Osula, who added a third in the 65th before a brilliant long-range consolation by Taty Castellanos.
Earlier, Bruno Fernandes had grabbed his record-equalling 20th assist of the Premier League season to help Manchester United beat Nottingham Forest 3-2 in an end-to-end match featuring the latest VAR controversy.
Bryan Mbeumo's 76th-minute goal proved to be decisive and was set up by Fernandes, who moved level with Arsenal great Thierry Henry (2002-03) and former Manchester City star Kevin De Bruyne (2019-20) for most assists in a single Premier League campaign.
The Portugal midfielder has one game left to set the outright record — at Brighton next weekend.
United had earlier gone in front for the second time in the game when Matheus Cunha stroked home a finish in the 55th minute after Mbeumo controlled the ball using his outstretched arm before having a shot blocked.
The goal was awarded by the on-field referee and he stuck to his decision — judging the contact accidental — after the VAR recommended he take a look on the pitchside monitor.
Luke Shaw gave United the lead in the fifth minute and Morato equalised in the 53rd. Cunha's contentious goal regained the lead for the hosts and Mbeumo made it 3-1, before Morgan Gibbs-White pulled another goal back for Forest.
In the race to qualify for Europe, seventh-place Brighton missed the chance to jump to sixth after losing 1-0 at Leeds, whose winner came from England striker Dominic Calvert-Lewin in stoppage time.
Eighth-place Brentford equalised late — through Dango Ouattara for his second of the match — in a 2-2 home draw with Crystal Palace to move one point behind Brighton. Ismaila Sarr and Adam Wharton, with his first for the club, netted for Palace.
Sunderland are a further point back in ninth after winning 3-1 at Everton, with three second-half goals.
The teams finishing in sixth and seventh place are set to qualify for the Europa League, with eighth place seemingly going into the Conference League.
15°C