Brown said it wasn't just about tourism, but also about the health of the island residents.
"Also for a humanitarian perspective, allowing travellers between our countries without the need for the 14 day supervised quarantine. That's having a big impact of people who need to go to New Zealand for medical services, elective surgeries, cancer check-ups and that sort of thing."
However, a statement from Jacinda Ardern's office suggests no timeline has been established, and details are still being worked out.
"The Prime Minister spoke to the Cook Islands Prime Minister yesterday to convey that she has asked officials to work on timelines for reopening with realm countries," a spokesperson for Ardern said.
"There are no set dates yet, and any speculation at this stage would be very premature. The number one priority will continue to be the safety of both New Zealand and Realm countries. No one wishes to be responsible for Covid entering into the Pacific."
Brown confirmed work was being done to ensure that passengers coming to and from the islands would not come into contact with travellers returning to New Zealand from other countries.
An Auckland Airport spokesperson told the Herald they remain committed to working with the Ministry of Health and government agencies to protect New Zealand against the spread of Covid-19.
"We are supportive of reopening the border for quarantine-free travel between New Zealand and other low-risk countries, when the Government decides it is safe to do so," she said.
"Auckland Airport stands ready to work with airlines and border agencies to ensure appropriate separation of passengers in the airport environment, including transit passengers; inbound passengers from countries who are part of a 'travel bubble' with New Zealand; and people arriving from higher-risk countries and going into 14-day quarantine.
"A range of operational, physical and time-separation measures will manage the segregation of passengers, with additional cleaning and sanitation measures also in place."
Air New Zealand also responded positively to the news of the Pacific discussions.
"Air New Zealand is pleased to hear discussions between the New Zealand and Cook Island governments on establishing a travel bubble have been going well. We look forward to providing safe travel between the two nations to more of our customers when possible."
The Cook Islands Tourism Corporation said the islands were desperate for quarantine-free travel between the two countries.
"New Zealanders comprise two-thirds of our international visitors - and the reality is they may well provide 100 per cent of our visitors for quite some time," said Graeme West, the agency's Australasian General Manager.
"With tourism responsible for over 70 per cent of the nation's GDP, our economy has been decimated since we closed our borders in March.
"We know there are many people keen for their annual getaway to the sun and I can assure them they would receive an especially warm Cook Islands welcome. Swimming, snorkelling and cocktails sound pretty good right now. The Cook Islands is proud to have remained entirely Covid-free and the Government will continue to take every precaution to ensure the safety of its people and visitors."
General manager of travel agency Our Pacific, Rick Felderhof, said a travel bubble between the two nations would reduce the Cook Islands' reliance on aid to get them through.
"Hallelujah, progress at last on international holidays to the Covid-19 free Cook Islands, where temps average between 25C and 29C and the locals are welcoming and friendly," Felderhof said in a statement to the Herald.
"The Cooks rely on tourism, and this exciting news will provide a 'tiare' branch for cash-strapped hotels, resorts, villas, and holiday homes, tour operators, and hospitality.
"The Cooks use the NZ dollar, and every dollar spent by Kiwi tourists reduces the need for the New Zealand Government to pump in millions of aid dollars to keep our realm country afloat."
