The Hammers sank to their 11th defeat of the season at the weekend as Arsenal comprehensively destroyed the bottom-placed side.
Beleaguered Grant appeared to signal his imminent departure by throwing his 'lucky' scarf to supporters after the game with Allardyce and Martin O'Neill hotly tipped to succeed him.
Going West? Avram Grant (left) could this week find himself ejected from the Upton Park hotseat and Sam Allardyce (right) has been linked with the job
It is understood that the Hammers board will convene for an emergency meeting this week - possibly as early as Monday - to discuss Grant's seemingly bleak future in charge of the club.
Allardyce, who is looking for a new challenge after he was shown the door at Blackburn Rovers in December, said he has had no contact from co-chairmen David Sullivan and David Gold but refused to rule out the prospect of an almost immediate return to Premier League management.
'There's been no word from West Ham,' he said. 'I will evaluate each and every offer as it comes and then meet whoever wants to take up my employment and take it from there.
'Avram has not been released from his position so it is not for me to speculate on that.'
Hammered: Grant's men were crushed at home by an Arsenal side inspired by Theo Walcott and Robin van Persie on Saturday
Allardyce would be an unpopular choice among the club's supporters who do not want a manager to comprise with their 'Academy' brand of football.
Former Aston Villa boss O'Neill remains the Hammers first choice and has been pressed to give them an answer over the next 48 hours after holding preliminary talks.
If O'Neill accepts West Ham's SOS, Sportsmail understands Tottenham's Robbie keane would be his transfer priority and Steve Walford would come in as his assistant.
But former Nottingham Forest team-mate John Robertson would not form part of the 58-year-old's backroom staff.
Furthermore, if the Ulsterman does replace Grant he is unlikely to be in charge by the time West Ham travel to Everton on Saturday.
Source: Daily Mail