The man was shot in Yuendumu, which is about 260 kilometres northwest of Alice Springs, on Saturday evening.
He was in critical condition last night, and died this morning.
The NT police force’s Acting Deputy Commissioner Michael White is expected to speak about the incident at 9am local time.
An elder in the Yuendumu community, Harry Nelson, told the ABC there were no medical staff there when the shooting occurred, as they had been evacuated last week after a series of break-ins. He said it took hours for an ambulance from a nearby community to arrive.
Mr Nelson said local residents had gathered outside the police station, where the man was being held last night, in an attempt to check on his condition.
“Police are not letting anyone see him and we don’t know how he is,” Mr Nelson said.