Eight men have held the office in the past.
Labour's foreign affairs spokesman David Shearer told RNZ National he thought Clark would "have a crack" at getting the job.
"The word on the street is that she's been for some time talking about it and testing out the waters with various people."
Prime Minister John Key has said he would support a bid by Clark, if she was to put her hand up for the job.
The term of Ban Ki-moon, current head of the world body, will finish at the end of next year, with the UN hoping to begin its selection process by the end of July next year.
Reforms within the UN's selection process will allow General Assembly member states for the first time to see basic information about all candidates, including their resumes, and to meet and question them.
But a number of barriers remain firmly in place, with the five veto-wielding permanent members of the Security Council: the United States, Britain, Russia, France and China, having the most say.
Herald political editor Audrey Young said Clark's chances rested on whether Eastern Europe could come up with a consensus candidate acceptable to Europe, the United States and Russia.
"With the current crisis in relations over Ukraine, that is not a simple task.
"If it can't find a consensus candidate, Clark will almost certainly seek the post and will be a front runner."
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A campaign for more transparency in the selection process, called 1 for 7 Billion, has been conducting an online poll asking what qualities people would like to see in the next UN chief.
"Have the courage to stand up to powerful countries" leads the way with 38 per cent of the more than 325 responses so far, spokeswoman Justine Brouillard said.
New Zealand was one of six elected security council members to sign a statement saying the council had been slow to engage with the General Assembly on the selection process, complaining that this is essential given the call "for dialogue and a more transparent and inclusive process of decision-making on this issue".
- Susan Strongman NZ Herald and AP