Australia’s second most populous state Victoria on Monday said it was investigating two suspected positive COVID-19 cases in Melbourne, which if confirmed would be the first community transmissions in the state in nearly three months.

Victoria was the worst hit Australian state during a second wave of COVID-19 late last year, accounting for about 70% of total cases and 90% of deaths. The state only controlled the outbreak after one of the world's longest and strictest lockdowns.

Australia has avoided the high COVID-19 numbers seen in many developed countries by closing its international borders in the early stages of the pandemic, lockdowns and social restrictions. It has reported just over 30,000 cases and 910 deaths.

The report of the new likely infections come as the federal government considers a plan that would allow fully vaccinated residents to travel freely between the states if regional borders have to be closed during future COVID-19 outbreaks.

During past outbreaks states have imposed internal border closures and mandated 14-day quarantine for visitors.

"As the vaccine rollout progresses, we can give greater certainty to Australians in terms of what being vaccinated delivers for them by way of benefits across the country," Finance Minister Simon Birmingham told ABC Radio on Monday.

Officials hope the vaccination and travel plan will prompt many Australians to get vaccinated. Australia's vaccination drive has missed initial dosage targets and lags many nations.

Many Australians are hesitant or delaying getting vaccinated because of the country's success in stamping out the coronavirus.

The vaccine rollout has ramped up in recent weeks with around a third of the 3.6 million total shots given in the last three weeks, but still far short of the 4 million pledged by March-end.

Prime Minister Scott Morrison last week suggested fully vaccinated travellers should be exempt from hard lockdowns and internal border controls but most states have dismissed the plan.

Morrison, however, will take his plan on travel exemptions to the national cabinet - the group of federal and state government leaders - next week, the Australian newspaper said in a report on Monday, without citing any source.