Thousands of Hammers supporters left London Stadium early for the second home game in a row as the Hammers were left a point above the bottom three.
"The pressure mounts game by game and at this moment it is big," Bilic, 49, told BBC Sport.
"We will see what the club will do. The club is above everyone."
West Ham have lost three of their past four games at London Stadium and were unable to cope with Liverpool's pace on the counter-attack throughout Saturday's game.
Mohamed Salah scored twice for the visitors, with Joel Matip and Alex Oxlade-Chamberlain also on target. Manuel Lanzini briefly gave West Ham home by making it 2-1.
Bilic said he would speak to co-owners David Sullivan and David Gold as usual after the defeat.
"I don't feel a broken man," he said. "On the other hand, the situation for West Ham is not good and the club is above every individual.
"I cannot say anything negative [about the owners]. They have been supportive, very good to me in my two and a half seasons, but the situation is very difficult."
West Ham's next match is at Watford on Sunday, 19 November.
"The international break does not change my situation, it makes no difference," Bilic said. "Of course I always believe in myself but it is not only about that now."
Former West Ham manager Sam Allardyce was a guest on Match of the Day on Saturday and he said "it wouldn't surprise" him if Bilic loses his job.
"There were rumours not so long ago that Slaven had two or three matches left and no contract extension pointed to the first 10 games being the most important," said the ex-England boss, who was in charge of the Hammers for four years before leaving "by mutual consent" in 2015.
"The owners have backed Slaven and spent a considerable amount of money.
"The players have lost their way because of constant changes. When you're searching for wins and draws you change the team. At this moment he looks so dejected.
"Structurally, they're not well coached."
'I never hide'
In his news conference after Saturday's game, it was jokingly suggested to Bilic that he should turn off his phone to avoid a call from Gold or Sullivan.
But the Croat said he would take responsibility for the club's poor form and was ready for the consequences.
"I will face the situation and the consequences," he said. "I never hide.
"I am the one taking the bullets all the time, when I do and don't have to. What is out of the question is how strong a character I am.
"But are we playing well? No we are not.
"No matter when it happens, sooner or later, if it's tonight, in one week, at the end of the season or in five years, I'm going to be very proud of what we've done in two and a half years."
Left-back Aaron Cresswell, exposed as the last man for the first and third Liverpool goals, said the squad were failing their manager.
"Are we letting the manager down? Definitely, every player included," he told Match of the Day.
"It is not good enough as a team collectively. We need to ease the pressure off the gaffer and ourselves.
"Confidence is low. It's not easy being at the bottom of the table. We saw the fans leaving at the end and that's down to the way we are playing. We need to turn this mess around."
Source: AFP
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