Party founder Bob Brown and veteran Senator Nick McKim are behind the move, co-sponsoring a motion at the Greens' conference this weekend in Tasmania.
If implemented, the shift to open tickets would hurt Anthony Albanese's party in marginal seats, given the Greens usually urge voters to preference Labor.
Brown, the 80-year-old environmentalist revered by party faithful, was blunt on his motivation.
"There is a two-word reason to support this motion: Murray Watt," he said.
Taking over as environment minister after Labor's election win in May, Senator Watt approved the North West Shelf gas extension, angering climate advocates.
As forestry and fisheries minister during the Albanese government's first term, he also drew the ire of environmentalists for his support of the salmon industry.
In response, the Queenslander said he was focused on environmental law reform and delivering housing and renewable energy projects.
"It's sad to see someone like Bob Brown, who has a long record of environmental campaigning, degenerating to personal attacks," he said.
Senator McKim said the Greens held broad concerns about Labor's policies.
"Environmental issues like forests and fish farms, and also on things like the AUKUS nuclear submarines, the relationship with the US under Donald Trump," he said.
"Labor and the Liberals are in zombie lockstep on many of the issues that matter dearly to our voters ... it's time for the Labor party to stop taking our preferences for granted."
The motion, if successful, would be put at the party's upcoming federal conference.
In practice, it would see the Greens move away from how-to-vote cards telling supporters how they should number every box, in favour of an open ticket.
Major parties typically produce these guides to voters following privately-negotiated preference deals, which can include bargaining over policies or support for other candidates.
"What we are simply saying is we trust our voters," Senator McKim said.
Senator Watt's office has been contacted for comment.
Preferences are an increasingly important factor in determining federal elections, as more Australians choose to vote for minor parties.
At the 2025 poll, just two-thirds of Australians gave their first preference to either Labor or the coalition - the lowest proportion on record.
Typically, Labor candidates win seats by benefiting from a strong majority of Greens preferences, while victorious coalition MPs are helped by preference flows from right-wing parties.
Mr Albanese's 2025 landslide win came off the back of Greens support, given Labor polled just 32 per cent of the primary vote: its third-lowest figure in almost a century.
Still, the impact of any Greens shift to open tickets may be minor: electoral analysts believe the majority of Greens voters will continue to preference Labor, even if undirected.
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