A senior figure at the club says the Hammers' hierarchy is undecided about whether Wednesday night's 5-0 defeat at Newcastle will be Grant's last game in charge.
But it is believed the board has no plans to hold an emergency board meeting on Thursday to discuss the manager's future.
Speculation surrounding Grant's position began to mount before Christmas with the Hammers having spent all season in the drop zone.
A four-match unbeaten run during the festive programme boosted their hopes of survival, temporarily lifting them out of the bottom three.
But a woeful performance on Tyneside, which saw the Hammers slip back to the bottom of the Premier League, has put the Israeli coach's position back under scrutiny.
Bookmakers have seen a steady stream of bets on Grant to become the next Premier League manager to lose his job with Sky Bet making him 5/4 second favourite behind Roy Hodgson to go.
Sky Bet's Dale Tempest said: "West Ham have been struggling since the very first day of the season and despite some recent better performances last night's 5-0 thrashing at Newcastle was probably the low point of the Hammers season. It seems a matter of "when" not "If" Grant is replaced