"We are reconsidering our position here," he said. "We'll be able to make a decision on when we can resume quarantine-free travel by Friday our time (Saturday NZT)."
But that doesn't mean a resumption of tourism just yet.
Talks were well underway for a travel bubble between the two countries in July, but the recent outbreak of coronavirus in Auckland burst that.
But with that outbreak seemingly contained, Brown and his New Zealand counterpart Jacinda Ardern have both said those talks have resumed.
Brown said things had progressed well and there were only a few finer points to iron out, which he hoped would only take a matter of weeks.
Those points included protocols for maritime borders and shipping, customs clearance, and how passengers to and from the Cook Islands would be separated from one another.
"Once that is done then both governments should be in a position to announce a date," he said.
"I am hopeful that December is a possible timeline, if we can get arrangements sooner then that would be a bonus."
Brown said there was significant demand, both from New Zealanders wanting a Cook Islands holiday, but also Cook Islanders wanting access to healthcare and family, among other reasons.
He said the Cook Islands would likely retain the need for all passengers to provide a negative Covid test and temperature scanning at airports.
Talks with Australia
Mark Brown said he was excited by Australia's announcement that New Zealanders would be able to travel to New South Wales and the Northern Territory without the need for quarantine
Cook Islanders are New Zealand citizens and carry New Zealand passports, and Brown said Cook Islanders would also be eligible for that entry.
"This can only progress the discussions that we're having currently," he said. "I do expect to have some discussions with Australia ... about the possibility of travel to Australia as well."
"It only takes a small number of Kiwis to come to the Cook Islands to make a big difference to our tourism industry."
- RNZ
