Both fighters are keen to return to the winner's circle, after contrasting losses to Britain's Dillian Whyte this year.
Browne, (26-1), was brutally knocked out during the sixth round in March, while Parker (26-2) lost by unanimous decision last month.
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Last year, Parker's manager, David Higgins, told RadioLIVE that he had communicated with Browne's camp and a deal looked likely, before an agreement with Anthony Joshua was struck.
"Lucas Browne's people are talking to us about a big trans-Tasman affair," he said.
"It would probably make sense in Auckland or New Zealand, because Joe has a better fan base in New Zealand than Browne has in Australia, so you could mount a big trans-Tasman clash."
But plenty has changed since, as Parker has lost his last two fights and, doesn’t hold the WBO belt anymore.
Parker also hasn't knocked anyone out since his win over Alexander Dimitrenko in 2016, so a trans-Tasman clash could be the one to rejuvenate the Aucklander’s career.
Speaking to World Boxing News about Parker's loss to Whyte, Browne said the 30-year-old Brit deserved to win the fight over the struggling Kiwi.
"I haven't watched the Whyte vs Parker fight in full, to be honest," he said. "I've seen clips and they it show that Whyte was getting better," he said.
"Whyte is one of the better fighters in the world at the moment, but I personally would love a chance to fight Parker and test my skills against him.
"From what I've seen, the fight was never a draw, and Whyte won it convincingly."
Browne, 39, has an upcoming fight scheduled against Kiwi-American Julius Long in Queensland for September 28, while Parker wants to have another fight before the end of the year.