Chief Minister Michael Gunner said food supplies to the community had been secured and the first batch of 71 tests arrived in Darwin on Sunday, and would be prioritised.
No one in that community had so far shown any symptoms of the disease and wastewater tests in surrounding communities, including Amanbidji, Bulla, Daguragu, Kalkaringi and Yuendumu, had returned negative results, he said.
Lajamanu has a low vaccination rate, with only about 35 per cent of residents double-dosed.
Omicron testing underway after passengers with COVID-19 arrive in Sydney from Africa
Acting Chief Health Officer Charles Pain said the positive wastewater test was a concern.
"If we have a wastewater positive, we expect to find a case in the community. It's been highly reliable," he said.
"There can be people who are shedding in the community and may not be actively infectious.
"We hope that will be the scenario."
The positive tests reported in the NT on Sunday included three linked to the Katherine cluster, which has now grown to 56 cases.
The fourth case is in a flight crew member.
All are in quarantine with the Katherine cases in isolation since before they became infectious.
Mr Gunner said the new cases and the situation in Lajamanu were evidence that "COVID is going nowhere".
"It will eventually end up everywhere. Every day we stop or slow the spread of COVID is a good day but it is hard work," he said.
"There is no single measure that can totally keep COVID out of your community."
Dr Pain said officials were still waiting on results from genomic sequencing in relation to a confirmed case in a traveller from South Africa to determine if they were infected with the heavily-mutated Omicron variant of the virus.
He said those test results might be available on Monday.
"It is a slightly low positive result, so we might have to do some extra testing, gathering more virus from the individual," he said.
"So we don't have that confirmation yet."
Urgent genomic sequencing is also underway in NSW to establish whether two people who tested positive on arrival in Sydney after spending time in southern Africa have the new Omicron strain.
In other states and territories on Sunday, Victoria posted 1,061 new locally acquired COVID-19 cases and four more virus-related deaths.
NSW reported 185 new cases and no deaths on Sunday, while the ACT reported seven new coronavirus cases.
In response to concerns over the Omicron variant, NSW, Victoria and the ACT will now make all overseas arrivals quarantine at home for 72 hours.
People already in the state who have been in the nine countries in the past two weeks must isolate for 14 days and be immediately tested.
Anyone in those jurisdictions who have been to the nine countries in the past 14 days must get a PCR test and quarantine immediately.
South Australia reported no new infections on Sunday.