Second choice to Darren Randolph last season, the 30-year-old Adrian has been relegated to a reserve role this campaign following the arrival of Hart on loan from Manchester City.
"He's working hard in training to get his chance and is getting his chance in the cup only, but that can change, of course," Bilic told the British media.
"I can understand players want to play and that's why they are sportsmen at the top level. But we also need a good squad and good keepers.
"Adrian is a great professional and is acting and behaving as I would expect... for the sake of the team, we need their professionalism and competition."
West Ham are 15th in the table with eight points from eight games and host Brighton and Hove Albion on Friday. The south coast club are a spot above their rivals with a similar record but a better goal difference.
(Reporting by Shrivathsa Sridhar in Bengaluru, editing by Pritha Sarkar)
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