New Zealand is unhappy about the boy's detention, accusing Australia of ignoring United Nations agreements.
The teen has been held at the Melbourne Immigration Transit Accommodation facility since March, awaiting a deportation hearing this month, reportedly over non-violent offending.
NZ Foreign Minister Winston Peters - standing in for Jacinda Ardern who is on maternity leave - has told reporters the issue has been raised with the Australian government.
"We are joint signatories, along with many other countries, to UN conventions where children are concerned," Mr Peters said on Tuesday.
"I'm just reminding the Australians, 'you're a signatory, live up to it'."
But Mr Dutton said the sooner the boy leaves Australia, the better.
"We don't want him here in Australia and if New Zealand want him back, then he's welcome to get on the first flight out," he told reporters in Sydney.
"We will make sure that he's deported at the first available opportunity but at the moment he's delaying his return to New Zealand."
More than 600 New Zealanders have had their visas cancelled in Australia since stricter deportation laws came into place in 2014. Some have spent the bulk of their lives in Australia and have no connections with New Zealand.
Mr Peters said he would be discussing the issue with counterpart Julie Bishop at a meeting in August.
"The Australians are very aware of what our views are on that and the expulsion of so many New Zealanders from Australia," Mr Peters said.
"We've made it clear for a very long time we don't like it ... but Australia is a sovereign nation."
The United Nations Convention on the Rights of the Child states "every child deprived of liberty shall be separated from adults unless it is considered in the child's best interest not to do so", except in exceptional circumstances.
