Dozens of former judges have heaped further pressure on Scott Morrison to establish a federal anti-corruption watchdog ahead of a crossbench push to set one up with or without government support.

In an open letter, 32 former judges have told Morrison a national integrity commission is needed to restore trust among Australians in the nation’s democracy.

Confidence in government and public institutions is at an “all time low”, according to the group, largely because of the suspicion that corruption permeates many government decisions and actions.

“Governments ignore at their peril demands by citizens to combat corruption with vigour,” they say in the letter published on Sunday. “We urge you to immediately establish a strong, effective and independent national integrity commission.”

The letter was coordinated by the Australia Institute, which has worked with legal experts including the former Transparency International chairman Anthony Whealy and the former Victorian court of appeal judge Stephen Charles to design a federal anti-corruption body.

It is also signed by the former chief justice of the high court Gerard Brennan and the first female justice of the high court, Mary Gaudron.

In January the Labor leader, Bill Shorten, sought to enlist the government in a bipartisan push to establish an anti-corruption body and promised to set one up, if elected.

The attorney general, Christian Porter, effectively rebuffed the offer in May, telling Labor there was no “persuasive evidence” that current methods of tackling corruption were insufficient.

The letter rejects that contention, arguing that existing federal integrity agencies “lack the necessary jurisdiction, powers and know-how to investigate properly the impartiality and bona fides of decisions made by, and conduct of, the federal government and public sector”.

It called for a body with “a broad jurisdiction and strong investigative powers, including public hearings”.

On Monday Shorten indicated Labor would seek support from the crossbench when parliament resumes this week, after crossbench MPs including Cathy McGowan signalled they would introduce a private member’s bill to establish an anti-corruption body.

“We will work with the Coalition but we will not wait for them,” Shorten said. “The people of Australia deserve nothing less.”

Porter responded to Labor’s latest push by saying it had failed to shed any light on how it wanted the watchdog to work. He said the Coalition was continuing to weigh up its options on strengthening the national integrity framework and was open to all policy options.

“As I have said repeatedly, I am not closed-minded as to options for a future integrity framework,” Porter said.