The internal fracture was labelled "astonishing" by Labor leader Bill Shorten, who accused the cabinet of leaking against itself after senior ministers lost each other's confidence.

It followed news that deputy Liberal leader Julie Bishop had proposed a "formal" inquiry into who leaked details of a spirited cabinet discussion in which Immigration Minister Peter Dutton suggested the government should consider dropping its opposition to Labor's policy of having the bank inquiry.

"First of all, she has confirmed that the cabinet is leaking on itself," Mr Shorten said.

"But what is more astonishing is that Julie Bishop has chosen to call for an investigation into her colleagues. The Turnbull cabinet is eating itself and Julie Bishop has said she doesn't have confidence in her cabinet colleagues."

Ms Bishop's remedy, driven by anger at being tagged as the leak herself, came as she took the unusual step of denying that it was her and as Treasurer Scott Morrison revealed he was to meet with bank chiefs on Thursday.

Sydney's Daily Telegraph reported elements of the Monday discussion in which Mr Morrison and Mr Turnbull were pressed to be pragmatic and consider an about face as Nationals prepare to cross the floor to help enable an inquiry.

Despite Mr Turnbull receiving support from former prime minister John Howard, who called a bank royal commission "rank socialism", neither the Prime Minister nor his disqualified deputy prime minister and Nationals leader Barnaby Joyce have managed to dissuade Coalition rebels, who argue the government should not have opposed the bank probe from the beginning - even if it is Labor policy.

"I would be staggered if the Coalition proposes a bank royal commission, that is rank socialism," Mr Howard told Sky News in an interview to mark the 10 year anniversary of his government's 2007 defeat.

"Our banks demonstrated in 2009 that they were amongst the best run, the most prudentially supervised and the most well-capitalised in the world."

But underscoring Mr Turnbull's woes, Mr Howard was not as complimentary in other areas, criticising the Prime Minister's refusal to insert religious exemptions into the same-sex marriage survey up-front, and criticising also the senior conservatives, Mr Dutton and Senator Cormann for not campaigning more vigorously for the "no" case.

Mr Turnbull received support from the banks on Thursday, with Westpac chairman Lindsay Maxsted warning it was "very dangerous" to send the global markets a signal that there were problems within the banking system.

"What messages does that send ... there's some problem there? They need a royal commission? What's that telling me that I don't know about? Very dangerous stuff," Mr Maxsted said.

Commonwealth Bank chief executive Ian Narev said lenders were already grappling with a suite of policy changes, tougher regulation, and court cases.

On another difficult day, Mr Turnbull was forced to field questions about the leak, but stopped short of committing to Ms Bishop's investigation.

"I don't comment on cabinet discussions let alone gossip about them," he said, flanked by Ms Bishop at the unveiling of the government's long-awaited foreign policy white paper.

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

Asked about the breach of cabinet confidence, Mr Morrison described it as "incredibly important that these issues are resolved", while Mr Dutton remarked that it made robust discussion difficult and reflected very poorly on the character of the minister involved.

Queensland LNP senator Barry O'Sullivan released an exposure draft of his bill to establish the banking probe on Thursday, setting the course for a showdown in Parliament before Christmas.

The government now faces the potential for a humiliating defeat on the floor of the House of Representatives - the chamber it is supposed to control.

The Queenslander is expected to gain the support of two LNP colleagues in the lower house, tipped to be fellow Queenslanders George Christensen and Llew O'Brien.

The Banking, Insurance, Superannuation and Financial Services Commission of Inquiry - which has the powers of a royal commission, but reports to the parliament, not the government - combines elements of maverick Queensland MP Bob Katter's bill, a separate Greens bill, and an additional focus on small business and rural lending.

The wide-ranging one-year inquiry would examine unethical or unlawful conduct by banks, insurance, superannuation and financial services companies, the regulatory system, agencies and laws that governs them, and whether a better dispute resolution system could be established to help consumers.

The Turnbull government argues it has already beefed up banking regulators and set out to establish better dispute resolution mechanisms for consumers.

Meanwhile, in a speech to the Business Council of Australia on Thursday evening, Finance Minister Mathias Cormann attempted to drag focus back to Mr Turnbull's speech earlier this week that foreshadowed personal income tax cuts in 2018.

Senator Cormann criticised Labor as the party of "higher taxes overall, higher taxes on business, higher taxes for savers and home owners" and pushed back at critics for suggesting the government was talking about tax cuts to distract from its current woes.

He pointed out the government had committed to a hard cap of tax as a share of GDP not exceeding 23.9 per cent.

This meant, effectively, "without future tax cuts, tax as a share of GDP would exceed 23.9 per cent".

Therefore, in the 2016-17 budget, "we have effectively already assumed future further tax cuts in our budget projections" - even if the company tax cut to 25 per cent for all companies was to be passed into law.