The Department of Defence confirmed the positive cases and close contacts were isolating as per COVID-safe protocols, adding the ship would continue on to Tonga and arrive off its coast early Wednesday morning.

HMAS Adelaide would fulfil its mission to support the relief effort, with humanitarian and medical supplies, engineering equipment and helicopters on board, the department said in a statement.

"Defence recognises the COVID-free status of Tonga, and will ensure the humanitarian supplies and equipment on board are delivered in a COVID-safe manner," it said.

"The ADF has significant experience conducting COVID-19 safe regional responses, including in Fiji following Tropical Cyclone Yasa, which will help contribute to the success and safety of this response."

HMAS Adelaide has "excellent" medical facilities on board and a 40-bed hospital.

All personnel are fully-vaccinated, with COVID-positive people either displaying mild symptoms or asymptomatic.

Earlier, Defence Minister Peter Dutton said the government was working with Tonga to ensure no threat to the Pacific nation.

"They need the aid desperately but they don't want the risk of COVID," the minister told Sky News.

"We will work through all of that as quickly as we can. We are not going to put the Tongan population at risk."

There are over 600 crew onboard.

It's the second aid shipment from Australia where a positive case has turned up, with a C-17 plane turned around mid-flight after someone was diagnosed with COVID-19.

Australian Associated Press