The Hammers have played three, lost three this season so far and it is believed the former Chelsea and Portsmouth chief is under intense pressure from co-owners David Gold and David Sullivan as a result.
The club has a string of tough fixtures in September and the Hammers board are expecting an immediate improvement and are preparing to review Grant's position if they remain bottom of the table.
They face Chelsea at Upton Park this weekend and then travel to Stoke and host Tottenham in their next three games.
Sullivan was reportedly furious with his team's lack of commitment during the 3-1 defeat against Bolton at Upton Park in August and would be willing to make tough decisions should supporter unrest continue and results don't improve.
Before the start of the season, optimism at Upton Park was high with the club expecting a mid-table finish but the terrible start has seen season expectations adjusted.
Grant did not play any of the club's new signings in the 3-0 surrender at Manchester United last weekend.