HARTFORD, Conn. — An offer from OxyContin maker Purdue Pharma and the Sackler family to settle some 2,000 lawsuits over their contribution to the national opioid crisis is receiving growing pushback from state and local officials who say the proposed deal doesn’t include enough money or accountability.

Connecticut Attorney General William Tong on Friday called for the company, which is headquartered in the state, to be forced out of the opioid business altogether.

“At a minimum, Connecticut demands that Purdue be broken up and shut down, and that its assets be liquidated,” Tong said in a statement.

He said he wants the controlling Sackler family to pay billions of dollars “they siphoned out of Purdue,” with the money going toward addiction research and treatment.

Massachusetts Attorney General Maura Healey said she wants any settlement to include more money than the $10 billion to $12 billion offered by Purdue and the $3 billion offered from the Sacklers, an amount that represents just a portion of the family’s fortune. Much of their money appears to be overseas.

“We owe it to families in Massachusetts and across the country to hold Purdue and the Sacklers accountable, ensure that the evidence of what they did is made public, and make them pay for the damage they have caused,” Healey said in a statement.

The company and the family did not answer questions Friday about criticism of the settlement proposal, under which Purdue would file for bankruptcy and transform itself into a “public benefit trust corporation.” The trust’s profits from drug sales would go to the plaintiffs under the company’s settlement offer.

Purdue also has been considering filing for bankruptcy protection on its own, an action that would upend the settlement talks involving state attorneys general and lawyers representing local governments around the country.

Over the past few years, nearly every state and about 2,000 local and tribal governments have sued over the toll of the opioid epidemic. Purdue is a defendant in most of the lawsuits and members of the Sackler family are named in several, including lawsuits filed by Connecticut, Massachusetts and Nevada in state courts.

The federal litigation is being overseen by a judge in Cleveland, who has been pushing for a national settlement but also has scheduled the first trial for October. Reaching a deal before then is proving difficult.

As details of Purdue’s settlement offer became public this week, some local government officials said it wasn’t good enough.

In Delbarton, West Virginia, Mayor Elmer Spence’s son and nephew are two of the more than 400,000 people in the U.S. who have died from opioid overdoses since 2000. If Purdue’s offer of a settlement is accepted, the town would receive less than $50,000.

“That’s a drop in the bucket for what it’s really cost this community,” he said of a crisis that has driven up costs for police, ambulances and courts.

He said residents in the town of roughly 500 people have grown accustomed to losing loved ones: “I mean it’s really a slap in the face.”

At the highest end of the proposed settlement payout, Ohio’s Jackson County could receive $2 million from the Purdue settlement, although the actual figure would likely be much lower. As high as that figure might sound, it wouldn’t begin to address the devastation of the epidemic in the county, said Robin Harris, who runs a tri-county addiction treatment office.

Officials have run out of foster homes, the county jail is out of bed space and treatment centers have long waiting lists.

“We’ve seen it devastate just the fabric of the community, starting with the family coming apart,” Harris said Friday.

Five years ago, the county’s human services agency had about 25 children in foster care, a number that has since doubled. Costs also doubled, with the small agency now spending more than $1 million a year to place children, many of them requiring expensive stays in residential treatment because of the trauma they’ve experienced, said Tammy Osborne-Smith, director of Jackson County Job and Family Services.

More than eight of every 10 custody cases involve a parent addicted to drugs, she said.

A settlement with Purdue would come in exchange for dropping the company and the Sacklers from lawsuits ahead of the October trial, which will examine claims from Ohio’s Summit and Cuyahoga counties. Other opioid manufacturers, distributors and pharmacies also are defendants in the federal litigation, but the negotiations with Purdue are further along.

The company has been strongly associated with the nation’s opioid crisis through its drug OxyContin and claims that it aggressively pushed its opioid painkillers while understating the risks of addiction.

The Sackler family has offered to give up control of Purdue and sell another company it owns, Mundipharma, contributing an expected $1.5 billion more to any settlement.

A person with knowledge of the negotiations who is not authorized to speak about them said at least some states are pushing for the family members to raise their contribution to $4.5 billion, in part because they don’t believe a sale of Mundipharma would generate as much money as the Sacklers estimate.

A court filing on Friday by the Oregon attorney general says the Sacklers engaged in a decade-long effort to siphon $11 billion out of Purdue.

Another source familiar with the talks said Purdue at one point in the negotiations had suggested paying a settlement that was partly in cash and partly in the overdose-reversing drug naloxone, but that was a nonstarter with the states. The company’s current offer includes money from profits of future drug sales, including naloxone.

A number of states are unhappy that settlement money would come from future profits of a company they blame for fanning the crisis. As the sides continue to negotiate, advocates for addicts and their families have started to push back and urge attorneys general to reject the Purdue offer.

Some want more money, while others want no settlement as a way to force the Sacklers and Purdue executives to answer for the company’s actions in court.

“We can’t let them get off the hook this easily,” said Sean O’Donnell of the Foundation for Recovery, a nonprofit in Nevada.