Prime Minister John Key met with Ms Clark, who is now the head of the UN Development Programme, in her office in New York across the road from the United Nations General Assembly hall.
After the formal meeting, at which Ms Clark said they discussed Fiji and Myanmar, the officials left the office so Mr Key could have a private talk with her.
Ms Clark said later they had "chewed the fat" over some issues - likely to be the Security Council bid.
Ms Clark can not openly lobby on the Security Council bid because of her current position. However, the campaign started when she was Prime Minister and her support for it is no secret.
"I've described the New Zealand bid as David against Goliath because it's up against Spain which is ten times as large as New Zealand in population and Turkey is 20 times as large. The point is David has been shown to win in other such contests, so anything is possible."
She said New Zealand was throwing everything at it, and had not served on it in 20 years.
"Truly New Zealand is seen as a good country around the UN. New Zealand doesn't have enemies - it is seen as a constructive and engaged, so there is always a lot of warmth for New Zealand."
Ms Clark also gave her take on Syria, saying she was "optimistic" that something would come out of Russia's proposal for Syria to hand over its chemical weapons.
Ms Clark said that it remained to be seen whether the Russian-Syria proposal would work if there were no sanctions on Syria if it did not hand over the weapons.
"But I think Russia now also has a big stake in the pathway that its promoted. So I'm optimistic something will come out of this."
She said the leaders' speeches at the General Assembly had made it clear there was not likely to be a military solution to the crisis. "I think people are getting a little depressed by the gridlock on the Security Council, so this is a new dynamic."
The Security Council is due to meet tomorrow and Syria will be on the agenda. There is some pressure on it to pass a resolution on Syria to require it to abide by the terms of the Russia proposal.