SYDNEY - A new travel bubble that allows New Zealanders to enter parts of Australia without quarantining has run into problems after arrivals quickly travelled to regions outside the bubble.

Australia's first travel bubble came into effect on Oct 16 after New South Wales, the most populous state, and the Northern Territory decided to allow people to visit from New Zealand without having to undergo the usual 14-day quarantine.

But the arrangement proved controversial after people began flying from New Zealand to Australia and then travelled on to other states.

The move angered state leaders, who said they had not agreed to allow New Zealanders to enter.

The Premier of Victoria, Mr Daniel Andrews, wrote to Prime Minister Scott Morrison to express his chagrin after 17 people landed in Sydney from New Zealand and promptly flew on to Melbourne. Victoria is emerging from one of the world's longest lockdowns as it tries to defeat a second wave of Covid-19 cases.

"We're disappointed this has happened," Mr Andrews said. "It is exactly the opposite of what we signed up for."

The federal government hit back, saying Victoria was aware of the arrangements with New Zealand and could have imposed measures to bar access to people from New Zealand.

Other states also expressed anger and moved to quarantine travellers who had entered from New Zealand.

The state of South Australia (SA) initially put about 12 arrivals from New Zealand on 14-day quarantine, but released them this week after assessing that they posed no risk. The state has now become the third territory in Australia to join the travel bubble.

New Zealand Prime Minister Jacinda Ardern has not agreed to waive quarantine requirements for those arriving from Australia, meaning the travel bubble is one-way only.

Australian state and federal leaders have urged Ms Ardern to let New Zealand join the travel bubble, though she has so far resisted. New Zealand is mostly free of community-transmitted Covid-19 cases after imposing stringent measures, including a strict lockdown.

South Australia's Premier, Mr Steven Marshall, said he has written to Ms Ardern to request reciprocal arrangements, noting that his state - unlike New South Wales - has no Covid-19 community transmission. He said Air New Zealand had indicated it could resume flights between Auckland and Adelaide, the state capital, within weeks if quarantine arrangements were lifted.

"There are still examples of community transmission in NSW; there aren't in SA, so I've laid out the case for SA having different arrangements to other parts of the country," Mr Marshall said.

Mr Morrison said on Friday (Oct 23) he believed the travel bubble with New Zealand was now working well, noting that he had just met with state and territory leaders and none had raised any concerns.

"I'm pleased with how that is proceeding," he told reporters. "There were no issues raised about that today, so that's good."

Mr Morrison said earlier this month that Australia would consider opening travel bubbles with Singapore, South Korea, Japan and Pacific island nations. He said he and Australia's Foreign Minister, Ms Marise Payne, had discussed safe travel arrangements with the leaders of countries across the region.

Earlier this month, Singapore began allowing travellers from Australia - aside those from Victoria - to enter without undergoing the usual quarantine requirements.

Qantas, Australia's national carrier, said on Friday it was prioritising resumption of flights to Asia amid hopes that travel bubbles could soon be established. But the airline said it did not expect normal travel to the United States or the United Kingdom to resume until a vaccine was in place because of the serious outbreaks in those countries.

"By early next year, we may find that (South) Korea, Taiwan and various islands in the Pacific are top Qantas destinations while we wait for our core international markets like the US and UK to re-open," the airline's chairman, Mr Richard Goyder, said at the company's annual general meeting.

Mr Morrison said on Friday he could not give a set a timetable for future travel bubbles.

"The quarantine issues… that's the first thing you've got to resolve," he said.

But he added: "We're already moving forward to try and solve these problems, so we can be proactive about this."