Andy Burnham, the shadow home secretary, is to call on the government to adopt a so-called “Hillsborough law”, that would give bereaved families the same resources as the police to make their case at future inquests.

The families of the 96 football fans who died in the Hillsborough stadium disaster had to raise the money to pay for their own lawyers at the earlier inquests in 1991, while South Yorkshire police had a top legal team.

Burnham, who campaigned on behalf of the families over many years, has tabled an amendment to the policing and crime bill, to be debated in the House of Commons on Monday, aimed at helping families in similar situations in future by granting them legal aid equal to the cost of police spending on lawyers. “The 27-year struggle of the Hillsborough families exposes how the odds are all too frequently stacked against ordinary families seeking truth and justice,” he said. “Never again should any bereaved families have to fight like the Hillsborough families have had to fight.”

Burnham plans to force a vote on the law. He will also advocate several other proposals, including urging the government to go ahead with the second phase of the Leveson inquiry into the media and restricting the ability of police officers to retire rather than face disciplinary action.

The home secretary, Theresa May, gave a statement in the House of Commons in April on the new Hillsborough inquests, including the jury’s finding that the victims were unlawfully killed in the crush at the stadium in 1989. She paid tribute to the families, saying: “They have faced hostility, opposition and obfuscation, and the authorities that should have been trusted have laid blame and tried to protect themselves instead of acting in the public interest. But the families have never faltered in their pursuit of the truth.”

The original Leveson report, published in 2012, examined phone-hacking cases and the nature of media regulation. But a second inquiry was envisaged, to investigate the relationship between the police and other public bodies, and the media. The government has said it wants to wait until all criminal investigations into phone hacking are complete before it commissions a new inquiry, but Burnham fears this is just a bid to kick the issue into the long grass. He would like to see the government pass legislation committing it to holding Leveson II.

Burnham said: “The government made a clear commitment to victims of press intrusion and parliament cannot let them renege on it. We need a full and thorough consideration of the relationship between press and police as recommended by Lord Leveson.”

Hacked Off, the campaign group representing the victims of phone hacking, is backing the call for a second inquiry, and lobbied MPs over the weekend to support the bid to force the government to legislate for one. A Home Office spokeswoman said she had no comment.

Burnham, who stood unsuccessfully for the Labour leadership last year, now hopes to be selected as the party’s candidate to contest the new metro mayoralty in Manchester. Elections to the new post, which will have enhanced powers over health, transport and economic policy as part of George Osborne’s ”northern powerhouse”, will be held next May.