Ninety minutes outside London, a turn down a narrow lane leads past fields of grazing cattle to a sign warning “Private Keep Off.” Around an elbow bend, a great stone manor, its formal gardens and tennis court hidden behind thick hedges, commands a 5,000-acre estate.

The estate is a pastoral prize — proof of the great wealth belonging to the family accused of playing a key role in triggering the U.S. opioid epidemic. But there’s little evidence of that connection. On paper, the land is owned by a handful of companies, most based in distant Bermuda, all controlled by an offshore trust.

The haziness surrounding the estate hints at one of the challenges for government lawyers as they eye a potential settlement with Purdue Pharma L.P. and its owners, the Sackler family, for their alleged role in flooding communities with prescription painkillers.

All but two U.S. states and 2,000 local governments have taken legal action against Purdue, other drugmakers and distributors. Sixteen states have sued family members by name, alleging they steered Purdue while draining more than $4 billion from the company since 2007. That’s when the Oxycontin maker pleaded guilty to misleading doctors, patients and regulators about the drug’s risks.

Purdue’s CEO has said the company could file for bankruptcy. And this week, news organizations reported that Purdue, the family and government lawyers are negotiating a possible settlement , valued at $10 billion to $12 billion, that would see the Sacklers give up company ownership and contribute $3 billion of their own money.

But where, exactly, did the money withdrawn from Purdue over the years end up? And how much might the family be holding that state and local governments should consider fair game?

Answers are complicated by the way the Sacklers have shielded their wealth in a web of companies and trusts, a review by The Associated Press has found. Some are registered in offshore tax havens far from Purdue’s Connecticut headquarters.

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Vermont Sen. and 2020 Democratic presidential candidate Bernie Sanders picked up the endorsement of the Pittsburgh-based United Electrical, Radio and Machine Workers of America on Monday, moments after calling for the working class to rise up and “tell the corporate elite that this country belongs to all of us” at the union’s annual convention Downtown.

Mr. Sanders, speaking to a few hundred union members at the Wyndham Grand Hotel, compared the union’s message to that of his own presidential campaign, saying that the U.S. cannot continue to be a nation in which there’s a “huge gap between the very rich and everybody else.”

“If there is going to be class struggle in the United States, it’s time that the working class got on the offensive and won that struggle,” Mr. Sanders said in an animated speech.

The UE, which represents about 35,000 workers in a variety of manufacturing and service-sector jobs, said they endorsed Mr. Sanders because of his consistent involvement in the union’s strikes and workplace actions.