Helen Clark has slipped from sixth to seventh place out of 11 candidates vying for leadership of the United Nations in the latest informal ballot, and she's now thought to be weighing her options.
Prime Minister John Key says the Ms Clark's chances are not looking as promising as they once did, but she still has his support.
"If they get a stronger leader, someone that's got the nimbleness, the dexterity to understand the issues, not an unguided missile, then you've got the best of both worlds - and in Helen you have someone who can understand those things," he told Paul Henry.
"Is she in the water? No. Is it an uphill battle from here? Absolutely. And what you've got is this block of votes that are piling up against her deliberately, because they want the rotation system to be preserved and they want an eastern European in."
He says one major challenge is not having a naturally big constituency.
"We found when we did the Security Council that if we looked at who would naturally vote for us because they're part of our region - Asia and the Pacific - our numbers were half what, say, the Europeans could rely on.
"It's a very weird system, because these straw polls don't of itself mean anything. They obviously are a way of trying to encourage some people to drop out of the raceâ€¦ That is a little different from when the permanent five members can exercise their veto."
The Security Council will hold secret ballots until a consensus is reached on a candidate to replace Ban Ki-moon, who will step down at the end of 2016 after serving two five-year terms.
Diplomats said the aim was for the council to recommend a candidate to the 193-member General Assembly for election in September or October.
The UN has never had a female Secretary-General, and in the lead-up to the straw polls there had been a major push for a woman to lead the organisation.