A collection agent for Medicare service providers is suing one of the dozens of trusts established by bankrupt asbestos manufacturers and plans to pursue others, alleging they have systematically shifted the burden of treating asbestos-related illnesses to the public.

MSP Recovery LLC said in court papers earlier this week it had identified 60 asbestos compensation trusts nationwide that fail to divulge settlement payments made to asbestos injury victims and to reimburse Medicare providers for those peoples’ healthcare costs.

Asbestos trusts established by bankruptcy courts have dispersed billions of dollars in recent decades to victims of mesothelioma and other asbestos-related cancers. MSP’s court filing targeted the trust established out of the 2002 bankruptcy of former industrial equipment maker J.T. Thorpe Inc., saying it and others have failed to notify or reimburse Medicare when paying compensation to people who were treated in the Medicare system.

The lawsuit, filed in the U.S. Bankruptcy Court in Los Angeles, said these bankruptcy trusts “routinely and consistently shirk reporting and reimbursement obligations under the law,” causing “staggering financial losses” for Medicare Advantage organizations, physicians’ groups and other healthcare payers.

The J.T. Thorpe trust’s executive director, Steven L. Bray, said it “believes the complaint filed by MSP Recovery and the related allegations are without merit and will vigorously defend itself.”

John Ruiz, the founder and chief executive of MSP Recovery, said that the company plans to bring five more lawsuits against asbestos trusts next week, and will have sued roughly sixty within the next two months.

Asbestos trusts have long been dogged by concerns from lawmakers and other authorities that they lack transparency or safeguards against fraud. Business interests have advocated for measures that would publicize details of claim payments, which trusts largely keep secret.

The lawsuit said the J.T. Thorpe trust administrators have a “continuing responsibility to provide notice of settlements with claimants who are Medicare beneficiaries, but failed, and continue to fail, to do so.”

MSP is poised to go public via a merger with special-purpose acquisition company Lionheart Acquisition Corp. II that would value the combined company at $32.6 billion. Its business is to sue over health expenses it contends should have been picked up by primary payers such as automobile, workers’ compensation or liability insurance companies but were instead covered by Medicare or Medicaid.

The lawsuit alleged that, as primary payers, asbestos trusts must report the compensation paid out to asbestos victims and reimburse Medicare providers when the recipients are Medicare beneficiaries.

In correspondence with MSP Recovery that was included in the lawsuit, the J.T. Thorpe trust said it wasn’t bound by Medicare registration and reporting obligations and that its confidential claim settlement amounts payments can’t be disclosed.

Under the Trump administration, the Justice Department asked bankruptcy court to more closely scrutinize asbestos trusts, urging transparency in the compensation of asbestos claims.

Since asbestos, a product mined from the earth and used in a variety of industrial and construction processes, was found in the 1960s to cause cancer, many companies that dealt in the substance have filed for bankruptcy, unable to afford legal settlements and jury awards.

The J.T. Thorpe trust, formed in 2006, has paid out more than $167 million on over 5,000 claims from its inception through the end of 2020, MSP said. MSP requested information regarding which of these claimants had their medical treatments provided by Medicare, but the trust refused to disclose the data.

MSP said its own investigation showed with 85% certainty that 284 claimants’ asbestos-related medical expenses appear to have been paid for by Medicare. The government should have been notified of the expenses and reimbursed for them but wasn’t, according to the complaint.

Write to Alexander Gladstone at alexander.gladstone@wsj.com