Malcolm Turnbull’s resistance to a royal commission into banks is flouting public opinion and will test his own backbenchers, who have their fingers on the electoral pulse on this hot-button issue.

An Essential poll out Tuesday found that 64% supported a royal commission with only 13% opposed. Support has risen five points since April, as the debate spearheaded by Labor has intensified.

Bill Shorten knows he is on a good thing, and he intends to stick to it. Labor is out daily pushing the case for a royal commission.

Turnbull’s attempt to counter with his recent plan to bring bank executives before a House of Representatives committee at least annually has been seen as little more than token.

Even in parliamentary terms it is limited. Nationals senator John Williams is critical of the lack of involvement of senators. The Senate committee system has done a good deal of investigation into the instances of rorts by the banks, and Williams says pointedly: “Involvement of senators [in quizzing the bank executives] would give better outcomes and better questions”.

Williams is one of the Coalition backbenchers publicly pressing for the government to do more than it has so far signalled. He and Queensland Liberal Warren Entsch are pushing the idea of a banking tribunal, through which people with grievances could seek justice.

Entsch says “some of these people have been battling for over a decade” with the banks, which have the capacity to throw a lot of money at lawyers, holding the strong advantage over victims who lack resources. A tribunal would put banks' behaviour in the spotlight by showcasing the examples of malpractice, Entsch argues.

Its operation, which he says should include enforceable decisions as well as fines and redress, would be a disincentive to bad behaviour. Those who could seek redress would include victims of both financial advice and lending rorts.

A parliamentary inquiry into impaired loans for small business and property development, which reported in May, found “there has been – albeit in a minority of cases – a persistent pattern of abuse of the almost complete asymmetry of power in the relationship between lender and borrower”. The committee recommended a process to examine contentious cases and determine restitution in circumstances of “unconscionable conduct”.

Entsch argues the establishment of a tribunal would lead to quicker results than a royal commission, while acting as a stick on the banks.

When the new parliament starts, Labor will put up motions calling for a royal commission. Only the government can set up such an inquiry but if a motion was passed in the Senate – which is likely – it would be embarrassing for Turnbull.

Williams, who has been a leading participant in the debate about banking ethics, has called consistently for a royal commission but wants it into white-collar crime more widely. He says, however, that depending on the terms of reference he would probably cross the floor on a motion for a royal commission into banking. “I don’t want to make a hypocrite of myself in my last three years [in the Senate],” he says, but predicts the numbers would be there without his vote.

Crucially, the government would have to make sure it kept its numbers intact in the House of Representatives, where it has a majority of only one. Presumably it would – the stakes would be too high for defections on such a high-profile test so early.

But the mood in the Coalition parties' ranks will be important in determining what extra action Turnbull will have to consider, short of a royal commission.