Hart has been told that he will be replaced by Adrián for the game against Chelsea this lunchtime and a continued absence would affect his World Cup chances.
"I don't want anyone to feel they have a divine right to play", Moyes said ahead of Saturday's home match against Chelsea.
Hart will be desperate to return to the side instead of Adrian with his 2018 World Cup hopes hanging in the balance if he fails to turn out for the Hammers on a regular basis.
But Moyes says Hart is going nowhere: "I don't think we could work here without two top goalkeepers".
"The problem is that I don't know if I have got someone who is that type of player to be a man-marker". I like us to be much better when we get the ball as well. "Hazard is a player who would definitely come into that category".More news: UN Envoy Meets N. Korean Foreign Minister
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Since the former Everton, Manchester United and Sunderland boss took over from Slaven Bilic, West Ham have picked up just one point from four matches, with just a draw against Leicester all they have to show for their efforts in that time. West Ham will likely concede at least once, they've only kept three clean sheets all season, so they'll need someone to create some scoring chances for them and Lanzini is their best bet.
But with Chelsea offering different problems, Moyes is ready to adapt to deal with the threats of Eden Hazard, Alvaro Morata and Cesc Fabregas.
"I would always try to speak to the players". You've heard the saying 'if he goes to the toilet, follow him.' I don't know if I've got that. I have to find the players who will help me win.
"It needs to be people who are athletic and quick". You say to them, "don't you think about doing anything else, this is your job", because he is that important.
'Looking through the squad we've got at the moment I couldn't really identify anyone who could do that but, tactically, it would be a good idea to stop him'.