Prime Minister Jacinda Ardern says the New Zealand woman identified as an global terrorist by Turkish authorities should have been deported to Australia.

New Zealand's Ministry of Foreign Affairs and Trade said it was aware of the information.

Jacinda Ardern said the woman was an Australia-New Zealand dual national, but that Australia's Government recently stripped her of citizenship under national security laws.

Ms Ardern accused Prime Minister Scott Morrison of "abdicating responsibility".

Ardern said they had "continually raised with Australia our view their decision was wrong".

"They left New Zealand at the age of six, were resident in Australia from that time, became an Australian citizen, left from Australia to Syria, and travelled on an Australian passport", she said.

Ardern said her government has an obligation to its citizens regardless of the circumstances or offences committed, and that decisions would be driven by the fact that two small children "who did not make the choice of being born in a war zone" were involved.

The 26-year-old woman was arrested with her two children near the S yrian border this week by Turkish authorities, and identified as a member of the Islamic State Group.

Ms Ardern's unusually strong remarks came after a woman and two children were reportedly picked up by Turkish officials for illegally entering from Syria. Later that year she was informed Australia had revoked the woman's citizenship.

"The welfare of the children also needs to be at the forefront in this situation".

"I think New Zealand, frankly, is exhausted of having Australia export its problems".

"If the shoe were on the other foot, we would take responsibility".

Mr Morrison, speaking at a press conference in Canberra an hour later, said it was his job as the Australian prime minister to "put Australia's national security interests first".

"The legislation that was passed through our parliament automatically cancels the citizenship of a dual citizen where they've been engaged in terrorist activities of this nature". Coming to New Zealand, where they have no immediate family, would not be in their best interests.

"What's on my mind is the concept of country's responsibilities for the people they have helped rear".

"These people who have grown up in Australia have networks and communities they can rely on to build support".

Ardern said it was wrong that New Zealand should shoulder the responsibility for a situation involving a woman who has not lived in New Zealand since she was six, has resided in Australia since that time, has her family in Australia and left for Syria from Australia on her Australian passport.

Reporters in the press gallery said the normally even-tempered leader was "visibly furious".

Danzeisen also highlighted the radicalisation of the woman, and the bumbling jihadi Mark Taylor - who spent years in Australia - and the Christchurch mosques' terrorist.

She said the work was important because it wasn't aimed just at cases such as the 26-year-old but others, including those who made threats to mosques.

It was work the Chief Coroner should undertake as part of the inquiry into the Christchurch shootings, she said.