The Hammers are sixth in the league table at the moment with 46 points from 28 matches.
The London outfit are just a point behind Manchester City, who are currently fourth in the standings and have a game in hand.
Bilic, though, is not getting carried away, and has made it clear that he and his players are taking it one game at a time.
“We are not thinking about the top four,” the West Ham boss told BBC Sport (22:19). “The gap might be small for those places but there is also a small gap between us and ninth place. We are going to continue to play like this and try to win every game.”
West Ham have been brilliant all season, and it would be great for the fans if they finish in the top four.
The way the Hammers played and won against London rivals Tottenham Hotspur at the Boleyn Ground on Wednesday evening suggests that they can make a push for a top-four finish.
Bilic’s side will return to action on Saturday afternoon when they take on Everton away from home at Goodison Park in the Premier League.