However, Slater and Gordon – and perhaps some of Australia's other leading class action law firms – are investigating Centrelink's handling of the scheme, and have not ruled out challenging the legality of its conduct under Commonwealth laws.

Slater and Gordon practice group leader Tim Finney told Fairfax Media that individuals could have grounds to appeal the validity and legality of debt notices sent by Centrelink.

"Slater and Gordon is currently reviewing Centrelink's conduct for the purpose of confirming whether it has engaged in any contraventions of applicable laws," Mr Finney said.

"Further, in some instances, the system that Centrelink has used to calculate the alleged debts is likely to have produced inaccurate debt notices.

"We are currently examining whether we may be able to provide affected individuals assistance in appealing debt notices they have received from Centrelink, on a case-by-case basis."

There had been widespread speculation one of the big law firms would sue the government for damages over its debt recovery program, but it's understood Slater and Gordon, Maurice Blackburn and Shine Lawyers have all ruled it out.

However, the Commonwealth could still face legal action for compensation, with at least one legal group investigating applying for compensation under federal laws.

Labor and the Greens announced on Wednesday they had secured support for a Senate inquiry into the data-matching system, which was designed to claw back millions in overpayments to welfare recipients.

The wide-ranging inquiry will examine the resources given to Centrelink to roll out the program, the role of staff, and how many notices were sent in error.

The Commonwealth Ombudsman launched a separate inquiry last month.

The debt-recovery process uses an automated system to match information held by Centrelink and the Australian Taxation Office and calculate overpayments.

Many of those targeted for debt recovery say they are being hounded by debt collectors, or threatened with jail, for money that they do not owe.

Victoria Legal Aid says the automated debt recovery system has boosted visits to the Centrelink information section of its website by more than 500 per cent since Christmas.

While it is considered unlikely a class action would succeed, Zach Banks – one of the leaders of community group Centrelink Class Action – said his fight for legal action against the government would continue.

"That robo-debt recovery is an absolute atrocity against the lowest-paid people we have," he said. "We live below the poverty line."

Mr Brady said that he had contacted police with serious concerns for the welfare of two members of the group who had threatened to take their own lives online, to draw attention to their plight.

"People who already have mental health issues are being put under the pump more and more and more, and it's not fair," he said.

Human Services Minister Alan Tudge told federal parliament on Tuesday the government was doing "important work" trying to get back money that people had rorted from the system.

"The reason we have to do this important work is because it is the unfortunate reality that some people deliberately misuse, abuse the system," he said.