Three-year-old Tharnicaa is being treated at Perth Children's Hospital after being evacuated from the island with her mother Priya Murugappan.

Her mother says Tharnicaa was sick for almost two weeks and that medical contractors at the immigration detention centre repeatedly refused to take her to hospital.

"I want to thank everybody for their love and good wishes," Priya said in a video message released on Tuesday.

"I hope that Tharnicaa can get the help she needs now. Please, help us to get her out of detention and home to Biloela."

Family supporter Angela Fredericks said Tharnicaa suffered dangerous temperature spikes on the journey to Perth and has since been diagnosed with a blood infection resulting from "untreated pneumonia".

"They are continuing to run tests, as they are still not able to get her white cell counts where they should be," she said.

"They are now treating the pneumonia while they look for any other infection sites."

The Department of Home Affairs and Australian Border Force said Tharnicaa had been receiving medical treatment and daily monitoring on Christmas Island consistent with medical advice.

"As soon as the ABF was advised by the treating medical practitioners that the minor required medical treatment in Western Australia, the minor was transferred to a hospital in Western Australia," an official said.

"The Australian Border Force strongly denies any allegations of inaction or mistreatment of individuals in its care."

Ms Fredericks said medical arrangements for detainees on Christmas Island, far off the coast of WA, were inadequate and dangerous.

"From my understanding, this would usually show up as a chest infection, which would then get treated," she said.

"And that would stop it going to pneumonia. If it went to pneumonia, that would then be treated to stop it going into the blood supply.

"We've had two delays in treatment here, which has led to this crisis point."

The department said detainees had access to dedicated nurses, doctors and referrals to specialists, and healthcare was "broadly comparable" with that on the mainland.

International Health and Medical Services (IHMS) is contracted to provide primary and mental health care services at the detention centre.

"Appropriate and timely medical care was provided by IHMS to the minor living on Christmas (Island)," the agency said in a statement.

"Due to privacy reasons we are unable to respond to specific questions relating to the medical treatment and condition of the individual."

IHMS did not say when Priya first asked for her daughter to be transferred to hospital, or how many times she asked.

Family supporter Bronwyn Dendle said IHMS had initially dismissed Tharnicaa's symptoms as the common cold, "despite persistent high temperatures and vomiting and diarrhoea".

Asked about the Tamil family on Tuesday, Home Affairs Minister Karen Andrews said "a range of resettlement options" were under negotiation.

"I can't make public commentary on that at the moment because I don't want to disrupt those negotiations," she told reporters.

Prime Minister Scott Morrison quickly added that she was not specifically referring to the Tamil family but "all cohorts, across all possible groups".

Mr Morrison said while the family's battle to stay in Australia remained before the courts, they would continue to be provided with "every medical care".

WA Premier Mark McGowan said Tharnicaa would remain in hospital in Perth until she recovered.

"I just urge (the federal government) to resolve the issues regarding this family as soon as possible," he said.

Priya and husband Nades settled in the Queensland town of Biloela after arriving separately by boat from Sri Lanka.

Tharnicaa and her sister Kopika, 6, were born in Australia.

The federal government has vowed to never permanently resettle anyone who arrives illegally by boat, and does not consider the girls to be Australian citizens.

The family has been in detention since 2018, and on Christmas Island since August 2019.

Labor Senator Kristina Keneally, who visited the family on the island earlier this year, has urged the government to let them return home.

Supporters will hold a candlelight vigil outside Perth Children's Hospital on Wednesday evening.

Australian Associated Press