Brenton Tarrant is Australian and had been living in Dunedin until his alleged killing spree at two Christchurch mosques on Friday.
Ardern was asked by reporters yesterday whether Tarrant was likely to be deported to Australia.
"I don't want to go to far down that track while we're obviously in early stages. Charges have been laid, we can expect additional charges, he'll be appearing in the High Court on the 5th of April, so there's obviously a process that needs to be gone through here.
"But I can say I am seeking advice on what will happen thereafter."
Asked today whether Ardern was referring to deportation before a sentence was served or after, a spokesman said she was looking at the issue in its entirety and getting advice on all options.
Tarrant did not require a visa to enter New Zealand because as an Australian he was able to enter the country and live here without one.
Ardern would not say how long Tarrant had been to New Zealand but said he had visited "sporadically".
At present Tarrant is charged with one count of murder under the Crimes Act.
Police Commissioner Mike Bush said that would change and police were considering all Acts it could lay charges under, including the Terrorism Suppression Act and more charges under the Crimes Act.
"We'll work towards the Crimes Act charges as a priority. I've already spoken about how many victims of these attacks there are, that will obviously present itself in the charges," Bush said today.
Tarrant's alleged crimes would qualify him for a sentence in life in prison if convicted.
Law Professor Andrew Geddis of Otago University was unsure whether New Zealand and Australia had an agreement under which prisoners convicted in one country could serve their sentence in their home country but said Australia would not want Tarrant back if he was convicted.
"I'm thinking it would require some kind of a one-off deal with Australia to allow that to happen," Geddis said.
