According to the Mirror, the Croatian's job remains on the line after the Hammers threw away two points against Crystal Palace at the weekend, as Wilfried Zaha's 96th minute goal rescued a 2-2 draw for the Eagles after West Ham had gone two goals up.
While the result meant the Hammers have lost just twice in the last nine games in all competitions, they have not won in their last three Premier League fixtures and remain just one point above the relegation zone.
With 96 minutes on the clock, West Ham winger Michail Antonio opted to fire a loose cross in the hands of Palace keeper Julian Speroni rather than hold onto the ball as the clock ticked towards full-time.
Antonio's mistake allowed Roy Hodgson's side one last chance, which Zaha seized upon to rescue a point for his team.
"We knew that the game was not over at 2-0 up at half-time," Bilic said after the game. "I am very disappointed with the penalty, but the way we conceded the leveller was very disappointing.
"We needed to keep the ball and we made the wrong decision, with Michail Antonio choosing to cross the ball rather than keep it. We have to manage the game on the pitch. The game was finished."
West Ham co-owners David Gold and David Sullivan were reportedly dismayed at the way the side relinquished a two goal lead at Selhurst Park and made their feelings known to Bilic when they met with him on Monday [30 October].
However, while West Ham's league position remains perilous, the club's board was encouraged by what they saw last week, when the Hammers produced an inspired comeback to overcome a two goal deficit and beat Spurs 3-2 at Wembley in the fourth round of the Carabao Cup.
The result proved Bilic retains the support of the dressing room and that the players remain committed to the cause, but the club's priority is to pull away from the relegation zone over the coming weeks.
West Ham desperately need points, as the first three fixtures in December could provide crucial for Bilic's future, as his side face Manchester City away, before hosting Chelsea and Arsenal in consecutive weeks at the London Stadium.