Moyes played Adrian in the 2-1 defeat at Manchester City last week because England star Hart was ineligible, and the Spaniard produced a fine display.
The Hammers manager is now set to keep Adrian in place for tomorrow’s clash against Chelsea at the London stadium.
But Moyes insists the move is not an indicator that Hart is dropped for good.
“It is a good problem for West Ham,” the Scot said.
“I don’t know if it is a good problem going into the World Cup.
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“I would like those problems in a few other positions.
“I am happy having the two of them.”
West Ham are seeking a first win under Moyes, having claimed just one point in four games since the former Manchester United manager took over.
The Hammers resolutely defended against City but were eventually beaten by a David Silva volley with seven minutes remaining at the Etihad.
Asked about the decision to retain Adrian in goal, Moyes said: “I wouldn’t give the team out because I don’t want to help Chelsea so I’ll keep my team until tomorrow.
“I’ve got my decision in my own mind but I’ll wait to announce it.
“I would always try to speak to the players, there are some occasions you should speak to them and others there is no need to.”