Seven months after the Supreme Court struck down a deal that would have resolved thousands of opioid cases against Purdue Pharma, the company’s owners, members of the Sackler family, have increased their cash offer to settle the litigation — but with a novel catch.

Under the framework for a new deal, the Sacklers would not receive immunity from future opioid lawsuits, a condition that they had long insisted upon but that the court ruled was impermissible.

Instead, they would pay up to $6.5 billion — $500 million more than the previous agreement — but with a new condition: Claimants, including states, municipalities and individuals, would have to set aside as much as $800 million in an account akin to a legal-defense fund for the billionaires to fight such cases, according to people familiar with the negotiations.

Some details of the framework — but not the legal-defense fund — were announced on Thursday by the New York attorney general, Letitia James. She said the overall settlement totaled $7.4 billion, which would include $897 million from Purdue.

New York could receive as much as $250 million, she said.

“The Sackler family relentlessly pursued profit at the expense of vulnerable patients and played a critical role in starting and fueling the opioid epidemic,” Ms. James said.

When the deal is finalized, she added, the Sacklers will “no longer have control of Purdue and will never be allowed to sell opioids in the United States again.”