The number of new cases in Australia had surged in recent months during the Southern Hemisphere’s winter, with the state of Victoria at the center of the outbreak. But officials there hope to be able to ease most restrictions over the coming weeks. This month, Australia began recording daily increases in the low double digits, down from more than 700 new daily infections at the peak of the outbreak.

New Zealand also had a spate of new cases in August, after a cluster emerged in Auckland, the country’s most populous city. Contact-tracing efforts appear to have prevented major community spread, allowing New Zealanders to return to a degree of normality.

If implemented, the Australia-New Zealand travel bubble would mirror approaches in Europe, where officials dropped some travel restrictions during the Northern Hemisphere’s summer but reimposed temporary measures for particularly virus-stricken cities, regions and, in some cases, countries. But New Zealand and Australia have set comparably tough thresholds, with New Zealand aiming to be essentially free of the virus, which will make the creation of a travel bubble more complicated than in Europe.

Top Australian politicians have publicly backed the idea of a travel bubble, despite lingering uncertainty about key details, including who would be eligible to travel and whether the arrangement would be reciprocal. According to Australia’s Nine News, only New Zealand residents from the country’s South Island — deemed to be virus-free at the moment — would be allowed to make the trip at first.

“Ultimately, whether New Zealand opens up to Australia will be a matter for New Zealand,” Australian Tourism Minister Simon Birmingham told Nine News on Sunday.

The New Zealand government cautioned that a travel bubble should be open only to residents of regions deemed sufficiently safe — an approach Australian lawmakers had objected to earlier in the pandemic, as it would mean that residents of some Australian states could go to New Zealand before parts of their own country.

“What we’d need to be assured of is when Australia is saying, ‘We’ve got a hot spot over here,’ that the border around that hot spot means [people] aren’t able to travel into the states we are engaging with,” Ardern told TVNZ.

Such an arrangement probably would prioritize travel from New Zealand to and from states such as Queensland, which has had only six covid-19 deaths since the pandemic began.

Meanwhile, Victoria, which has had 787 deaths, would be likely to remain off-limits.

For the first time in nearly two months, residents of the state’s capital, Melbourne, did not have to hurry home on Monday night to make curfew.

Australia’s second-largest city, the epicenter of the country’s largest coronavirus outbreak, had been under severe restrictions since early July.

With infection numbers plummeting, Premier Daniel Andrews announced Sunday that the surrounding state of Victoria had hit its benchmarks “ahead of schedule.” Although some restrictions will stay in place, child-care centers can begin reopening, and outdoor public gatherings will be allowed for up to five people from no more than two households.

“We have made more progress than we hoped to make at this point in time,” Andrews said Sunday, while cautioning that the virus “will run wild” if all precautions are abandoned.

Australian Prime Minister Scott Morrison and others had argued in favor of an even faster easing of restrictions in the state, echoing concerns about the economic and mental health effects of prolonged curbs.

The state’s tourism industry, which accounts for about 110,000 jobs, has been hit particularly hard. Despite the announcement of a $177 million infrastructure and tourism stimulus package last week, a significant number of jobs are expected to disappear over the coming months, with lingering uncertainty about when international tourism will resume.

But farther south, on the Australian island state of Tasmania, the potential arrivals of New Zealand visitors within weeks was greeted with enthusiasm on Monday.

“New Zealand’s a sleeping giant market for us, potentially,” Luke Martin, the chief executive of Tasmania’s Tourism Industry Council, told the Australian Broadcasting Corp.

Antonia Noori Farzan contributed to this report.