"The Islamic community and all of New Zealand has already suffered enough without having to pay astronomical prison costs to keep him safe in our prison system."
It has been estimated that keeping Tarrant in a New Zealand jail for the rest of his life would cost millions of dollars.
Prime Minister Scott Morrison said Ms Ardern had not raised the suggestion of a prisoner transfer with him.
"It's normal practice that criminals convicted of these offences serve their sentences in that jurisdiction,and that's my understanding of what the arrangements are and no request has been made to Australia for that to be any different,"he said.
"Justice today was delivered in New Zealand to the terrorist and murderer for his cowardly and horrific crimes and attacks on a Christchurch mosque. The world must never see from,of him,or hear from him ever again.
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"All Australians were and remain horrified and devastated by his despicable terrorist act. New Zealand is family to us in Australia. Today,we send our love across the ditch."
International prisoner transfers are a matter for Attorney-General Christian Porter,not the Home Affairs Minister.
Mr Porter's spokesman said that under the International Transfer of Prisoners Act 1997,the Australian government could only transfer prisoners from a country recognised as a"transfer country"under the ITP Act.
"A transfer country is a country which either has a bilateral agreement or arrangement for prisoner transfer with Australia,or is a party to a multilateral agreement or arrangement for prisoner transfer such as the Council of Europe Convention on the Transfer of Sentenced Persons.
"New Zealand is not a transfer country under the ITP Act as it does not have any agreement or arrangement for prisoner transfers with Australia,"he said.
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In handing down Tarrant's sentence,Justice Cameron Mander said there was no minimum period of imprisonment to sufficiently denounce this crime.
"Your crimes are so wicked that even if you are detained until you die,it would not exhaust the requirements of punishment and denunciation."
The killings were"brutal and beyond callous".
"Your actions were inhumane ... You showed no mercy. You are not only a murderer,but a terrorist. You sought to essentially attack New Zealand's way of life,"Justice Mander said.
Ahead of the sentencing,91 victims and their family members addressed the court about the impact of the brutal massacre perpetrated by Tarrant.
Ms Ardern has been critical in the past of Australia deporting New Zealanders who have been convicted of a crime,once they have served their sentence,and called for the policy to be rethought.