Foreign Minister Julie Bishop's calls for an investigation into Cabinet leaks is indicative of a government "eating itself", according to Opposition Leader Bill Shorten.

Key points: Julie Bishop calls for an investigation into Cabinet leaks

Julie Bishop calls for an investigation into Cabinet leaks PM says he takes security of discussions "very, very seriously"

PM says he takes security of discussions "very, very seriously" Bill Shorten says the leak shows the Government is "eating itself"

Ms Bishop said she supported a formal investigation, after reports in the Daily Telegraph earlier this week suggesting Cabinet had discussed scrapping its fervent opposition to a banking royal commission.

She said a strong message needed to be sent that leaking is "unacceptable", and reminded colleagues it was a criminal offence.

But this evening she told the ABC's 7.30 program it was up to Prime Minister Malcolm Turnbull to decide whether to hold an investigation.

"That's a matter for the Prime Minister," she said.

"Successive governments have held formal investigations.

"I recall the Labor Government having an Federal Police inquiry into leaks."

Mr Shorten argued that was an "astonishing" admission by the deputy Liberal leader.

"But what is even more astonishing is that Julie Bishop has chosen to call for an investigation into her colleagues," he said.

"The Turnbull Cabinet is eating itself, and Julie Bishop has said she doesn't have confidence in her Cabinet colleagues.

"So why should Australians have confidence in the Turnbull Cabinet?"

Mr Turnbull expressly avoided discussing the leak.

"I don't comment on Cabinet discussions, let alone gossip about them," he said.

"But I can assure you, I take the security of Cabinet discussions very, very seriously indeed."

Dutton had 'gut feeling' he knew who leaker was

Senior Cabinet ministers have also voiced their concerns that confidential conversations have made their way into the public spotlight.

"I think it's incredibly important that these issues are resolved, and that's a matter I'm sure the Prime Minister has well in hand," Treasurer Scott Morrison said.

But the most stinging criticism came from Immigration Minister Peter Dutton, who claimed he had "a gut feeling" about who the leaker might be, but said he would leave it to that colleague "to out themselves".

"If people are leaking, they don't do anybody any good, there's no good that comes of it," Mr Dutton told 2GB.

"If people believe that somehow their careers are furthered by leaking or by trying to put out something that's going to damage the Government, or damage me, or damage Malcolm, it says more about them than it does about us.

"If you've got leaks out of Cabinet, it's a poor reflection on the character of that person."

Mr Dutton argued there was a strong need for the sanctity of Cabinet discussions to be preserved.

"I offer my frank advice, I argue with the Prime Minister, he argues with me behind the scenes about different policies or decisions that the Government might make," he said.

"We don't live in a dictatorship, prime ministers want frank advice and I give frank advice."

Ms Bishop insisted the majority of Coalition MPs are behind Mr Turnbull, who she said had shown "extraordinary leadership" in recent months.