Peter Dutton has referred to the two children of the Biloela Tamil family as “anchor babies” and blamed them for costing taxpayers millions of dollars in incendiary comments defending the government’s decision to deport the family.

In an interview on Thursday the home affairs minister echoed anti-immigrant rhetoric from the United States, borrowing a term used by Donald Trump to justify a plan to end birth-right citizenship, to claim Labor, the Greens and refugee activists are attempting to “bully” the Morrison government into allowing the family to stay.

Labor has already emphatically rejected the label, arguing that it wants Dutton to allow Priya, Nadesalingam and their two Australian-born children Kopika, 4, and Tharunicaa, 2, to remain in Australia because they have been accepted by the community of Biloela in regional Queensland.

Dutton told 2GB radio the family’s case has cost taxpayers “literally millions of dollars” because refugee advocates are “armed with pro bono lawyers” who were able to get a federal court injunction while the plane had stopped in Darwin en route to Sri Lanka.

“It’s frustrating because … it’s been very clear to them at every turn that they were not going to stay in Australia, and they still had children. We see that overseas in other countries, anchor babies, so-called: the emotion of trying to leverage a migration outcome based on the children.

“I regret to say I don’t think this will be dealt with quickly. I think it will go on now for potentially a couple of months because lawyers will try and delay, and that is part of a tactic. They think if they delay they can keep the pressure on the government and we’ll change our mind in relation to this case.”

The case is listed at the federal court in Melbourne on Wednesday. Dutton claimed the case was being used by people smugglers to promote boat journeys to Australia, and targeted Labor for alleged inconsistency by seeking discretion in this case but not 6,000 other asylum seeker families in Australia.

“I don’t know if you say yes to this family how you can say no to the 6,000 families behind,” he said. “As soon as you allow those people smugglers to get back into control, the families and faces you won’t see will be [those of] the people that go to the bottom of the ocean again.”

Last week the opposition spokeswoman on home affairs, Kristina Keneally, pre-empted the Coalition’s new rhetoric, accusing the government of importing an “American debate about so-called anchor babies” due to Dutton’s frequent references to the parents choosing to have children in Australia.

But Keneally told Radio National: “The law is very different in the United States where citizenship is accorded to anybody born on American soil.

“That is not the law in Australia … The issue here … is that the Biloela community, Australians, have embraced this family, want them to be part of their community, have integrated them into the fabric of their community. It’s not simply the act of having a child.”

In March Dutton claimed accepting refugees and asylum seekers would result in the displacement of Australians seeking medical care, claims rejected by doctors administering new medical evacuation laws the Coalition is attempting to repeal.

The prime minister, Scott Morrison, backed Dutton over the comments, which he said were based on “simple math”, then later that month urged Australians to “reject the thinking that one person’s gain is another’s loss” in the aftermath of the Christchurch terrorist attack. Instead Morrison suggested Australians should “disagree better”.

On Friday the former prime minister Tony Abbott praised the far-right prime minister of Hungary, Viktor Orbán, and warned a conference in Europe about “military age” male immigrants “swarming” the continent.