United have just 10 points after seven Premier League games —that meagre haul plus their negative goal difference means this is their poorest start since 1989-90, when they had picked up seven points at this stage of the season.
United, who finished second last season behind Manchester City, already appear to be out of the title race and problems on the pitch are compounded by a growing rift between the manager and Paul Pogba.
West Ham made a mockery of their appalling record against the 20-times English champions at the London Stadium, with an early strike from Felipe Anderson and a Victor Lindelof own goal shortly before half-time leaving the disjointed visitors with a mountain to climb.
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Mourinho blamed the officials for West Ham's first and third goals, saying the opener should have been flagged offside and the move that led to the Arnautovic goal included a foul on Rashford.
"We needed a good start and we didn't have that good start. After the result on Tuesday (defeat to Derby in the League Cup) the team obviously needs positive things, positive feelings and to start losing after five minutes is not that positive start that you need."
The United boss defended a tactical switch to bring in Scott McTominay alongside Chris Smalling and Lindelof in a back three and the decision to play Anthony Martial instead of the misfiring Alexis Sanchez, who was not even in the squad.
The United boss brought off Pogba — to a chorus of jeers from the home fans — and Martial, with Fred and Juan Mata sent on to perform a rescue act.
All eyes at the London Stadium were on Pogba after the French World Cup winner was stripped of the vice-captaincy before a League Cup defeat to Championship side Derby.
The following day Pogba was involved in a terse exchange with Mourinho on the training pitch, further evidence that the relationship between the pair is at breaking point.
The win for West Ham is a further sign that Manuel Pellegrini's side have turned the corner after they started the season with four straight defeats in the Premier League.
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Former Manchester United defender Rio Ferdinand said the club faced some "big decisions" over manager Jose Mourinho's future after a damaging 3-1 loss away to West Ham on Saturday.
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United's third defeat in seven Premier League games left them eight points behind leaders Liverpool and three adrift of the top four, with Mourinho's men now having played one game more than the teams above them in the table.
"As a footballer your DNA is about hard work and effort, I didn't see that today," said Ferdinand, now a pundit with BT Sport.
Victory saw West Ham continue to climb up the table after starting the season with four straight league defeats under new manager Manuel Pellegrini.
Arnautovic was in no doubt the Hammers had earned their victory.
"It doesn't matter how they set up, our game was to try to win, we have to do our job and we did it excellently, and that is why we deserved to win," he said.
As for Mourinho, Arnautovic added: "I have nothing to say. I look at my team, whatever the other team is doing is not our stuff."
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