Labour's Palmerston North MP Iain Lees-Galloway has taken over responsibility for the End of Life Choice Bill after sponsor Maryan Street failed to get re-elected in September.
Mr Lees-Galloway is gauging support within the new Parliament before deciding whether to put it back in the private members' bill ballot.
Ms Street removed the bill from the ballot last year under pressure from Labour's leadership, who were concerned it could be an election-year distraction or that it could deter conservative voters.
Whether the bill gets a second chance could depend on Labour's next leader.
The four contenders have their last meeting in South Auckland tonight. The new leader will be decided on November 18.
Mr Little said he supported legalising euthanasia but he would rather the bill was not returned to the ballot.
"The challenge for the next three years is for us to emphasise issues of priority to a broad cross-section of New Zealanders and I'm not sure [euthanasia] is one of them.
"For a party that's at 25 per cent in the polls, where there is a clear issue about the level of trust and confidence in us to lead and be in government, this is not a priority issue."
Grant Robertson, who also supported legalisation, said he had advised Mr Lees-Galloway to establish a cross-party group in support of euthanasia to "take the issue forward".
"From a party point of view we've got a lot of other priority issues to sort out," Mr Robertson said.
Nanaia Mahuta backed her colleague to put the bill in the ballot, saying it would show that Labour "would stand up for those difficult conversations that need to be had".
David Parker did not want to comment until he had discussed the bill with Mr Lees-Galloway. He voted against a previous attempt to legalise euthanasia in 2003.
Mr Lees-Galloway said there was no certainty he would put the bill in the ballot until he had canvassed support in Parliament.
He said the worst thing for the issue would be for the bill to be heavily defeated at the first reading.
He emphasised it was an individual, not a party, issue and it would not distract from Labour's focus on bigger issues.
By Isaac Davison of the New Zealand Herald
