Two days of Federal Court hearings over the removal of Priya and Nades Murugappan and their Australian-born daughters Kopika and Tharunicaa, aged four and two, have finished in Melbourne.

No date has been set for Justice Mark Moshinsky to hand down his decision on whether the family will be allowed to leave Christmas Island and return to the Queensland community of Biloela.

"It's difficult to estimate as to how long it will be - potentially between one to three months, that's not unusual in cases like this," their lawyer Carina Ford said after the hearing on Tuesday.

"It really is one of those cases that could go either way."

The family's lawyer Angel Aleksov argued both Immigration Minister David Coleman and Home Affairs Minister Peter Dutton took procedural steps which required a decision on granting a visa for the youngest family member, Tharunicaa. But no decision had been made.

Maritime arrivals are barred from applying for visas while in Australia and, despite being born in Australia, the young girl was given the same status as her parents.

Mr Dutton lifted the bar for 7500 arrivals in July 2017, which Mr Aleksov said included Tharunicaa as an individual.

However, that has been challenged by government lawyers, arguing the then-one-month-old was part of her mother's application.

Mr Aleksov also claimed Mr Coleman received and considered two separate briefings from the department of Home Affairs, prepared in April and May last year. The latter included "options" available to him, including granting various types of visas.

The May briefing was returned to the department with the notation "not for action".

Mr Coleman's senior adviser Ross McDonald said he couldn't recall seeing a briefing, despite records showing he was given a hard copy, or giving it to the minister.

But he had "no comment" on whether Mr Coleman had asked him for a briefing.

Mr Coleman was not called as a witness himself. He has been on leave for unknown personal reasons since December.

Mr Dutton, the court heard, received a briefing from department secretary John Pezzulo in July 2019 which included considering the government's international protection obligations in relation to the family.

Mr Aleksov said there was no point to consider these options if the minister had not agreed to take the formal procedural step of considering a protection visa.

Stephen Lloyd SC, representing the ministers, rejected claims Mr Coleman was responsible for decisions relating to the family until July last year, when Mr Dutton took over, saying both had responsibility for different aspects.

He said if Mr Coleman had requested information on the family he was saying "tell me what I could do", rather than taking a formal procedural step.

The protection assessment was a step taken by the department and not Mr Dutton, he added.

Justice Moshinsky will hand down his decision at a later date.

Australian Associated Press