Senior Coalition ministers have continued to dig in on their hardline stance on the Tamil family detained on Christmas Island, despite mounting community calls to show compassion.

The family’s youngest daughter Tharnicaa, who turns four in just days, remains in a Perth hospital after being evacuated for medical treatment earlier this week.

The Morrison government has been under mounting pressure for its treatment of the family after supporters revealed the three-year-old was refused medical attention for days before her emergency hospitalisation.

A wave of public support has gathered momentum this week, with candle-light vigils in Perth, Sydney and Canberra.

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More than 500,000 people have signed a change.org petition calling for the federal government to allow the family to return to Biloela, Queensland, where they lived before being taken into detention in 2018.

On Friday, Social Services Minister Anne Ruston deferred questions about the family to the immigration and home affairs ministers.

But she also seemed to suggest showing Priya and Nades Murugappan and their two young girls leniency would restart the “disgusting sights” of the people smuggling.

“The Australian government takes our responsibility for the care of anybody that is in our care very, very seriously,” Senator Ruston told ABC radio on Friday.

“But we equally take the security of our border very seriously.”

Attorney-General Michaelia Cash has also warned of the “consequences of blinking” on border security.

Tharnicaa is said to be in a stable condition and responding well to treatment for a blood infection believed to have been cause by untreated pneumonia.

On Thursday, supporters of the family said doctors were waiting for further test results and the youngster would remain in hospital for several days.

“This also means that Tharni will be in hospital for her 4th birthday on Saturday,” her supporters wrote.

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The family has been detained for almost three years as the federal government attempts to deport them from Australia.

Priya and Nades Murugappan met and were married in Australia after they separately fled Sri Lanka’s civil war by boat in 2012 and 2013.

Tharnicaa and her older sister, six-year-old Kopika, were born in Australia, while the couple lived in the central Queensland town of Biloela.

The rest of the family have exhausted their legal options.

On Thursday, a suggestion from Foreign Minister Marise Payne that the family could be resettled overseas was vetoed by Home Affairs Minister Karen Andrews.

Because the family has no refugee status, they had not access to agreements with New Zealand or the US, Ms Andrews said.

This week, the Minister’s office was sending all calls about the family to voicemail, with this message:  “If you wish to raise a query in relation to the Muruguppan Sri Lankan family, please send through your concerns in writing via the ‘contact the minister’ webform on the home affairs website,” it said.

Some Liberal Party backbenchers have also reportedly privately petitioned for the family to be reunited on the mainland.

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Some state leaders have also joined calls for their release. Queensland Premier Annastacia Palaszczuk added her voice on Friday.

“What’s happening to that family is incredibly tragic. They were a family based in Biloela and it’s time the federal government made a decision,” Ms Palaszczuk said.

On Thursday, WA Premier Mark McGowan said the federal government needed to “sort it out”.

“If that means they need to use one of their exemption powers under the Act, just do it and resolve the issue – we actually need workers in Australia at the moment – and get this very unfortunate and somewhat internationally embarrassing issue past us,” he said.

Labor MPs have also been prominent in calling for the foursome to be released.“Karen Andrews could just make a decision to release this family,” frontbencher Tanya Plibersek said this week.

Labor’s Home Affairs spokeswoman Kristina Keneally, who recently travelled to Christmas Island, claims the it has cost nearly $50 million to detain the family on the island. Official estimates put the total cost at more than $6.7 million by May 2021.

-with AAP