The east London club find themselves just two places above the bottom three after a dismal run of form, including a 3-0 defeat to Burnley on Saturday.
But it is understood that those in charge at the London Stadium are willing to give the Chilean boss more time to turn things around, having allowed previous manager Slaven Bilic and David Moyes to see out their contracts before letting them go.
You can certainly see both sides to this one. On the one hand, Pellegrini is stirring the Hammers into some pretty choppy waters at the moment, and while it was only a few short weeks ago that they were being touted as European football contenders, now they look more like relegation fodder. That is simply not good enough, and there has to be some kind of accountability held. On the other hand, however, the Hammers have invested a lot of money into the Pellegrini tenure, and to kick him out now would look like an admission that they have wanted their money. On the flip side, there can be little doubting Pellegrini’s pedigree as a manager, and the hope must be that he can turn things around sooner rather than later. They are still only four points off fifth, and if they can pick up a couple of improbably wins over the next few weeks, perhaps this whole blip will just get forgotten about. That’s certainly what the board will be hoping anyways.