The Northern Territory has detected 12 new COVID-19 cases as health authorities investigate a mystery case in Darwin.

It was diagnosed overnight with NT Health saying little except the source of infection is under investigation.

Two of the other new infections are linked to the current community outbreak, bringing the cluster centred around Katherine, 320km south of Darwin, to 146 cases.

Both are people from the Tennant Creek area and household contacts of previous cases, NT Health said on Monday.

They were not infectious in the community and are in quarantine.

The other nine new cases include a close contact of two workers from Yulara who acquired their infections in Queensland.

They were not infectious in the community and are in quarantine at The Centre for National Resilience in Howard Springs, near Darwin.

Five cases are interstate arrivals and are all in isolation at home or at the Howard Springs facility.

Three cases are close contacts of interstate arrivals.

There are currently 15 people with COVID-19 in hospital, including one in intensive care.

The majority of admissions are for the purposes of infection control and assessment.

The Nhulunbuy community, 1000km east of Darwin, was encouraged to get tested for the virus after positive wastewater results were detected.

Meanwhile, police are searching for two men who failed to enter the Alice Springs Quarantine Facility after being directed to do so.

Akiir Lueth, 29, and Pashaiy Ayiik, 31, crossed into the territory through the Kulgera check point near the border with South Australia.

Police believe the pair are in Alice Springs after finding their red Ford Focus in the town.