Sportsmail understands Hammers chiefs are close to pulling the trigger after the humiliating 5-0 defeat at Newcastle on Wednesday night.
Grant is not the only manager in serious danger of losing his job. Liverpool's Roy Hodgson, Aston Villa's Gerard Houllier and Chelsea's Carlo Ancelotti are all fearing the worst after poor losses.
Firing line: Avram Grant, Gerard Houllier, Carlo Ancelotti and Roy Hodgson are all fighting to save their jobs
Despite a recent mini-revival prior to the St James' Park mauling, Grant now finds his side back on the bottom of the table and facing an uphill battle to stay in the top-flight.
West Ham co-owners David Sullivan and David Gold will now decide whether it's time to boot out the man they handed a four-year contract to only last summer.
Next up for West Ham is an FA Cup tie at home to Barnsley on Saturday but Sullivan and Gold could dump Grant before then.
A hat-trick from Leon Best - plus goals from Kevin Nolan and Peter Lovenkrands - inflicted a devastating defeat on the lowly Hammers to send them spiralling back down to the foot of the table.
The Israeli insisted: 'We were in a worse position one month ago. It's where we are at the end of the season that matters.'
Net pains: West Ham are all at sea as Leon Best scores his second goal of the night
Grant was not the only boss to suffer on Wednesday night. Liverpool manager Roy Hodgson is now clinging to his job after losing 3-1 to Blackburn at Ewood Park.
Hodgson said later: 'My job? I'm not prepared to talk about that, I am depressed enough about the performance.'
Asked about his feelings after the defeat Hodgson said: 'Dejection, bitter disappointment, frustration. I am very disappointed for the fans who gave us good support and saw the team concede the goals we did.
'I thought in the second half we did well to retrieve a seemingly hopeless situation and get back on track and had Steven's penalty gone in who knows, we may have made the last five minutes more exciting.
Another setback: Martin Olsson fires past Pepe Reina in Blackburn's win at Ewood Park
'Unfortunately when you are Liverpool and you lose away to Blackburn there is nothing that can be said which will put a positive slant or make the situation look any brighter.
'As far as I am concerned it is a really bad day and there is not much more I can say other than, to the fans, that anything they are feeling tonight I can assure them the feeling is equally as bad and equally as depressed in the dressing room.'
Defending champions Chelsea suffered another big blow to their hopes of retaining the title when they were beaten 1-0 by Wolves, who started the night bottom of the league.
But Ancelotti insists he remains the right man to lead Chelsea out of their current slump. He said: 'I don't fear (for my job). I'm worried because the team doesn't reach the result. I think it's obvious we are not good enough in this moment.
Night to forget: John Terry gets an earful from Wolves fans during their shock defeat
'But the team has the skill, the ability, the character, the personality to move on. It's important to stay compact, to stay together and to work hard for each other.'
Houllier is also facing increasing scrutiny after Villa dropped into the relegation zone after losing 1-0 at home to Sunderland.
Phil Bardsley scored his first top flight goal to leave the Holte End chanting towards Houllier 'you're getting sacked in the morning'.
Yet, Houllier insists he retains the support of owner Randy Lerner and chief executive Paul Faulkner despite his woeful start at Villa Park. It is understood that he is safe until at least the derby with Birmingham on January 16.
Piling on the misery: Kieran Richardson and Titus Bramble celebrates Sunderland's win
He said: 'I know about the chants. I heard that. It hurts but I understand their frustration. They were fantastic at the end and the players can't complain about the fans.
'Every time we were under the cosh, or every time we had to come back into the game, they were pushing and supporting the team. What is important is that you have the backing of the chairman and the chief executive.
'I spoke to Paul Faulkner and he was extremely supportive. I understand it is not easy for them. I don't doubt Randy Lerner's firm support and determination. We are in this together for better or worse. At the moment, we are going through the worse.'
Sack race! Time surely up for Roy Hodgson and Avram Grant after humiliating defeatsWatch all the Premier League goals every week on our brilliant video playerNewcastle 5 West Ham 0: Best leaves hammered Grant fighting for survivalBlackburn 3 Liverpool 1: Hodgson is deep fried by the Venky boysAston Villa 0 Sunderland 1: Bard to worse for under-fire boss HoullierWolves 1 Chelsea 0: Ancelotti future in grave doubt after Bosingwa own goal inflicts yet more damage to title defence
Explore more:People: Kevin Nolan, David Gold, Roy Hodgson, Gerard Houllier, Leon Best, David Sullivan, Carlo Ancelotti Places: Newcastle, Liverpool, Birmingham