The former Prime Minister refused to be drawn on whether that meant some other candidates were weak.
But both she and Key implied that there were candidates who would not be up to the job of handling some of the world's most difficult problems.
"We can see by the crisis that has been playing out in Syria how important it is to have a strong Secretary-General," Key said. "The United Nations needs to step up and pick a candidate that can actually do the right thing by the organisation, by the countries that they represent."
That is the message Key and Clark are likely to pursue in separate meetings with dozens of leaders throughout the coming week.
Key is due to meet British Prime Minister Theresa May early today (NZ time). He has formal bilateral talks with six other leaders but will be rubbing shoulders with many others at events over the next three days.
That includes two high-level summits on refugees and migration, one hosted by UN Secretary-General Ban Ki Moon and the other hosted by US President Barack Obama.
Key said "raw politics" of the situation meant countries were protecting the process of regional rotation.
He said the simple reality was that if New Zealand were part of Eastern Europe, Helen Clark would be the next UN Secretary-General.
"[It] is not a situation where other countries don't like Helen or don't respect her capability."
Clark agreed: "Nobody disputes that I can do this job but then you come into the geo-political factors and then is it a time to stray from the traditional diplomatic low-profile appointee to someone of a stature that I as a former Prime Minister and senior level leader here represent?"
She said the final decision would come down to a compromise and her aim was to still be in contest and an available option at that time - early October - after the last indicative ballot in a week's time.
"The strategy had been to stay in this race and when there's the shoot-out at the OK Corral to be standing then as an option," she said.
Clark said she was deeply appreciative of the support she had had from Key, Foreign Minister Murray McCully, the Ministry of Foreign Affairs and Trade and the New Zealand public.
UN Q&A
What is UN Leaders' Week?
The biggest annual gathering of leaders anywhere in the world, although attendance is staggered over a week. They are not all in the same room at the same time.
Do all leaders come?
No. This year, for instance Russian President Vladimir Putin will be represented by Foreign Minister Sergey Lavrov. But new British Prime Minister Theresa May will be attending as will President Francois Hollande of France and President Barack Obama. China is represented by Premier Li Keqiang.
What do they do?
Each of the 193 leaders or their nominees make a speech to the UN General Assembly and they hold private meetings on the sidelines.
Is there a theme?
Leaders can choose to concentrate on any subject but this year both UN Secretary General Ban Ki Moon and Obama are holding events on refugees and migration.
How many times has John Key attended Leaders' Week?
This will be the fifth time in the almost eight years he has been Prime Minister.
