Mr Key said he had not received any advice of her intentions, and it would be a hard job for her to get.
"It would be well and truly sought after and these things are deeply political. But she's done a very good job as the administrator of the UNDP. We would back her, but whether or not she can actually get there, I don't know."
A spokesman later said that if Helen Clark did approach the Government, it would look at the kind of support it could provide at that point, as it had when she ran for her UNDP role. However, it was too early to talk about specifics.
Labour leader David Cunliffe said through a spokesman that he also knew nothing of Helen Clark's intentions. "But we would hope that if it did become a live issue, the New Zealand Government would support such a bid."
The post is elected by UN member states. Labour MP Phil Goff, a former Foreign Minister, said support from New Zealand was likely to include lobbying on Helen Clark's behalf by the Prime Minister and government ministers meeting with their international counterparts. It could also mean contributing financially to her campaign such as through travel and accommodation costs.
Governments had done that for previous high-level international appointments, such as Sir Don McKinnon's role as head of the Commonwealth, Trade Minister Tim Groser's failed bid for the World Trade Organisation and Mike Moore's leadership of the World Trade Organisation. But Mr Goff said Helen Clark already travelled a lot in her present job and might not need such assistance. He said the networks she had built up in her time as Prime Minister and at the UN would work in her favour, as could coming from a small country.
He said having her in the post would be important for New Zealand's interests, and her experience meant she was more than capable of doing the job.