The former Everton and Manchester United boss has guided West Ham to fifth in the table this season and into an unlikely race for Champions League qualification. However a 1-0 defeat at Man United and a 3-3 draw with Arsenal (after the Hammers had led 3-0) have dented those chances.
Moyes is still widely viewed to have done a tremendous job - but it's not enough for one West Ham fan, who told talkSPORT that he was "livid" with the dropped points against Arsenal. So much so, he wants the manager sacked.
"Not really, no," he replied when it was suggested Moyes had done an amazing job to get West Ham fifth after 29 games. "Everyone's having a nightmare, aren't they?" he added, referring to the number of Premier League big guns that are struggling.
"It's not taking into account Liverpool's 'mare, Tottenham's 'mare, Arsenal's 'mare," he explained on why fifth was not good enough.
"This is our chance! Our chance to get top four - and if we had a Scott Parker or Brendan Rodgers as manager, I think we'd get there.
"And we'd get their manager. We're a big London dog."
Asked if he wanted Moyes gone, the caller responded: "Yeah."
While Moyes isn't always the most adventurous coach, it's hard to imagine any manager coming in and doing a better job at the club.
Asked to arrive and steady a relegation-threatened Irons - for a second time - Moyes recused West Ham last season. This campaign has seen the Scot take the club above the likes of Liverpool, Everton, Arsenal and Tottenham.
Admittedly, Rodgers is doing an excellent job at third-place Leicester. But there is absolutely no sign that he'd leave the Foxes for West Ham, who haven't had as much success as Leicester in recent years.
Parker is also doing well to turn Fulham's season around, but the club are still 18th and in the relegation zone. Which is exactly the place Moyes has lead West Ham away from.
But it seems even that can't please the most demanding of Hammers supporters.
All imagery: PA Images
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