Grant has been told he must win at least one of his next three games - against Blackburn, Fulham or Everton - to stand a chance of avoiding the axe.
Burned: Sam Allardyce is flying off to Middle East as soon as his compensation case is over but West Ham want to talk to him
The fact that West Ham's board are seeking talks with Allardyce shows that contingency plans are in place should Grant fail to meet his target.
Down and out: Glum Grant has three games to save his job
If he succeeds, Grant will be handed a transfer kitty to bring in players who will pitch into West Ham's battle against the drop.
The Israeli is keen on Tottenham midfielder Jamie O'Hara, who starred for him on loan at Portsmouth, and he wants a striker with a solid goalscoring record in the Barclays Premier League.
If Grant has not improved by the time West Ham play Wolves on New Year's Day, he will be sent packing and the transfer budget handed to his successor.
The first test for Grant comes coincidentally against Allardyce's former club at Ewood Park on Saturday.
And what makes Grant's position even more precarious is that West Ham's co-owners David Sullivan and David Gold, along with the club's vice-chairman Karren Brady, may have to move fast to persuade Allardyce that his future lies at Upton Park.
The former Bolton and Newcastle manager is hoping to wrap up his Blackburn compensation claim swiftly, before jetting off on holiday to Dubai next week.
Allardyce intends to remain in the luxury holiday destination until the new year. He may even find himself in demand for a lucrative managerial post in the Middle East.
Allardyce spoke to the United Arab Emirates League in the summer and only their failure to agree a compensation package with Blackburn prevented the veteran boss making the switch.
In demand: West ham are looking at signing Tottenham's Jamie O'Hara, who impressed on loan at Portsmouth last season
Former Leeds and Aston Villa manager David O'Leary is feeling the heat at Dubai club Al Ahli, with the club currently fifth in the table and nine points off the top.
Hen-picked: Rovers chairwoman in fresh attack on sacked AllardyceUnder-fire Grant wants to link up with Pompey hero O'Hara at West Ham Grant tells West Ham: No more excuses - just start winning pointsWest Ham, Everton and Fulham ready to fight for wantaway StranzlWEST HAM UNITED FC