Acting Prime Minister Michael McCormack has rejected federal Liberal MPs' calls for a Tamil family detained on Christmas Island to return home to Biloela in Queensland.
Two federal government ministers had voiced support for the family, saying it's time they were brought back to the mainland.
Indicating a possible shift inside government on the fate of the Murugappan family, Trent Zimmerman said that while the case was "very complicated", support for them in the Australian community was "overwhelming".
Immigration minister Alex Hawke will consider an application in coming weeks to use his powers to give an exemption to a Tamil family detained on Christmas Island, he told ABC News on Saturday.
"This week or in the next couple of weeks, (he) will be considering an application to use his powers to give an exemption to the normal requirements," Mr Zimmerman said.
"I've spoken to him and others and I hope he looks favourably on that request."
"I think its time we brought this to an end and brought them back to the mainland."
Mr McCormack disagreed with Mr Zimmerman.
"What we don't want to see is the boats return," he said.
"Trent was not in parliament when some of those ships were lost at sea, some of those leaky boats were dashed up against rocks and all lives lost.
"I was. I remember the heartache, I remember the loss."
LNP Queensland MP Ken O'Dowd, whose seat takes in Biloela, has reportedly also spoken out in favour of bringing the family home.
He called Alex Hawke on Friday morning to urge him to let them settle in Australia, telling the Nine newspapers that the immigration minister agreed with him that the matter had gone on too long and had to be resolved.
More than 500,000 people have signed a petition asking the federal government to allow the Murugappan family to return home to Biloela, Queensland, where they previously lived.
The family's younger daughter Tharnicaa, who turned four on Saturday, remains in a Perth hospital after being evacuated from Christmas Island for medical treatment earlier this week.
Mr Zimmerman's comments mark a departure from senior government ministers' consistent refusal to soften their hardline stance.
Social Services Minister Anne Ruston suggested on Friday that showing the family leniency would restart the "disgusting sights" seen at the height of the people smuggling trade.
Attorney-General Michaelia Cash has warned of the "consequences of blinking" on border security.
Home Affairs Minister Karen Andrews floated the possibility of resettling the family in the US or New Zealand before walking back the idea.
Mr Zimmerman said the family's case was complicated because they had been found not to be refugees and that Australia had already sent 1500 other Tamil families back to Sri Lanka.
Queensland Labor MP Anika Wells welcomed a possible shift in the government's stance.
"It's overdue time to bring them home," she told ABC News.
Supporters of the family believe the conditions in immigration detention may have contributed to Tharnicaa's illness.
Priya and Nades Murugappan met after fleeing Sri Lanka's civil war by boat in 2012 and 2013.
Tharnicaa and her older sister Kopika were both born in Australia after the couple established themselves in Biloela.
The family has been detained for almost three years as the federal government attempts to deport them from Australia.
Their deportation is being fought in the courts.
Mr Murgappan and Kopika sent Tharnicaa a video birthday message from Christmas on Saturday.
"Happy Birthday Tharnica. I miss you, I love you Tharnica, bye, see you," they said.
Another birthday video for Tharnicaa was circulated earlier on Saturday featuring Australian celebrities singing Twinkle Twinkle Little Star.
Australian Associated Press