West Ham are 16th in the table with eight points from nine games but Bilic said he would not walk away from the challenge.
"You ask me about Koeman, there are always three, four, five managers in those positions and I'm definitely one of them," Bilic said ahead of the League Cup fourth round clash at Tottenham Hotspur on Wednesday.
"Does it make me lose my focus? No, it gives you even more. I never gave up in my career, no matter in which way of life, as a parent, as a friend, as a manager.
"I am never, never, never walking away... It is my job and a job I like to do especially in a place that I feel, after my hometown, is like few other places, you are treated special."
British media reports said West Ham's owners have given Bilic the next two games to turn things around and save his job but the 49-year-old is not worried.
"That's my biggest strength, to be strong in hard times. Now I'm having a hard time, yes I am. That doesn't scare me."
After the clash against Tottenham at Wembley, West Ham travel to basement side Crystal Palace in the league. Palace have just one win in nine games so far.
(Reporting by Shrivathsa Sridhar in Bengaluru; editing by Sudipto Ganguly)
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