"This is about authority and whether she can assert her authority because she hasn't got it now," one senior party figure was quoted as saying.
The minority coalition government's popularity is at record lows in opinion polls, with an unpopular tax on carbon pollution adding to its woes, but Gillard insisted she remained the best person for the job.
"I'm not going anywhere," she told ABC radio.
"I'm the best person to do this job. And I'll continue to do it. And what this job is about is leading the nation to a better future."
Gillard said her government had taken some "tough" and "bold" decisions and several cabinet colleagues quickly gave her their backing, including Trade Minister Craig Emerson who described her as "a strong leader".
"They (Australians) expect leaders to make tough decisions even if those decisions aren't always popular in the short term," he said.
The government had heralded the Malaysia policy as a sure-fire strategy to deal with the politically divisive asylum-seeker issue and it is currently taking legal advice on what to do next.
The nation's top court found that under Australian law the government could not send asylum-seekers to be processed in a country that was not bound to adequately protect them.
Malaysia is not a signatory to the UN convention on refugees.
- AAP
