A 1-0 victory over Tottenham on Wednesday has left the sixth-placed Hammers just one point adrift of the Champions League spots with Saturday's visit to Everton the first of their remaining 10 matches this season.
Bilic admits the club's position in the table justifies talk of claiming a coveted berth in the top four, but is also cautious over setting his sights too high.
"It is realistic now. We are one point behind with 10 games to go so we have a chance," Bilic said.
"But it's so tight and there are still a lot of games to go. Of course we are looking above us because we are very close to those places.
"We are also looking below us because there are teams who are queuing up. The gap is not big between us and ninth or 10th in the table."
Source: PA