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But he also played down the prospects of her winning it.
"Those jobs are Herculean tasks to win and there's so much politics at play that it's not straightforward, but in the event that she ran, we would definitely support her," Mr Key told the Herald.
Helen Clark, the former prime minister whom Mr Key beat in 2008, is now head of the United Nations Development Programme (UNDP), which means she's No 3 at the UN.
She has not declared for the top job and is unlikely to do so unless she has substantial backing, but in an interview with the BBC's Hardtalk programme on Friday, she fuelled speculation that she is looking for support.
There were three main issues to decide, she said: did the countries of the United Nations want the position determined on a regional basis, did they want the traditional diplomatic style, and was it time for the first woman to lead the organisation.
It was an attempt to encourage debate about the importance of promoting the gender issue over the traditional approach.
Mr Key thinks the gender issue is important. "It's high time that the UN had a woman as Secretary-General.
"There is no question she is a hugely talented politician and she is uniquely placed, given her leadership of the UNDP," he said.
"I think it would be a very proud day for New Zealand if she became the next Secretary-General."
It is common wisdom at the UN that if the decision is made on a regional basis, the next Secretary-General would come from Eastern Europe.
A strong Eastern European woman who had the support of key powers on the Security Council could trump Helen Clark.
The term of Ban Ki-moon, a South Korean, expires at the end of 2016.
Mr Key is due to meet Helen Clark in New York on Wednesday morning (NZ time) while he is at the UN to lobby for NZ to get a seat on the Security Council in 2015 and 2016.
- Audrey Young in New York