Reports on Monday morning claimed Bilic was set for crisis talks with the Hammers hierarchy following Saturday's 5-1 demolition by Arsenal.
But Press Association Sport understands no such talks have taken place, other than Bilic's usual discussions with Sullivan after a match.
Bilic's post-match verdict on his players was damning, accusing some of them of a lack of dedication in training this season.
And Sullivan, speaking to the club's website, said: " I saw Slaven's comments after the game and as always he was completely honest with his assessment
"Slaven cares passionately about the football club and this defeat will be hurting him as much as anyone.
"I have no doubts that he is doing everything he can to address the situation and everyone is working together to ensure we turn our season around.
"We cannot forget the amazing job that Slaven did in his first season at the club
With a bit more luck he could have taken us into the top four.
"His passion, commitment and outstanding track record at the highest level were among the many reasons we appointed Slaven in the summer of 2015.
"Despite what some people have said, there is still a great spirit among the players and everyone is working towards the same objective."
West Ham have won just three Premier League matches all season and face a tough trip to Liverpool next weekend.
Source : PA