The Hammers are hovering three points above the relegation zone after a dismal run of seven matches without a win in the Premier League, including five defeats.
But the Mail Online reports that Pellegrini can still save his job by sparking a turnaround in their next two outings, away to Chelsea on Saturday and away at Wolves the following Wednesday.
Sources say that a drop into the relegation zone would spell the end for the former Manchester City boss, who has another 18 months left to run on his deal at the London Stadium.
Sacking the 66-year-old would leave the club with a compensation bill of around £6.5 million, which has increased their reluctance to make a hasty decision.
The West Ham hierarchy are also said to be sympathetic about the loss of goalkeeper Lukasz Fabianski to injury, as his stand-in Roberto Jimenez has failed to impress, leading to questions being asked of director of football Mario Husillos.
But the Premier League club are already searching for potential successors, despite handing the Chilean an ultimatum.
Bournemouth boss Eddie Howe, Burnley’s Sean Dyche and former Hammers manager David Moyes are the club’s three preferred options to take over.
They could face competition from fellow strugglers Everton for all three, as Marco Silva teeters on the brink at Goodison Park.
Howe and Dyche would be pricier options to go for, given that they are currently under contract.
In addition to their poor league form, West Ham were humiliated by League One side Oxford United with a 4-0 defeat in the Carabao Cup back in September.
That result kick-started the Hammers’ winless run after a promising start to the campaign.
Before their cup exit, Pellegrini had led his side to three wins in their first six games to put West Ham in the mix for the European places.
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