County Executive Edward Mangano has proposed a change to the jail medical vendor’s contract to keep the embattled company in place for three more months — with a significant raise — while working to cut a deal with Nassau University Medical Center to take over inmate health services.

The administration’s pitch would give Armor Correctional Health Services $4.8 million to stay past its May 31 contract end and remain at the East Meadow jail until Aug. 31.

“We don’t have another health care provider we can bring in for three months. We can’t have a break in health care for one hour,” Deputy County Executive Charles Ribando said Tuesday.

While Ribando said Armor’s contract is worth about $11.6 million a year now — or $2.9 million for three months — the change would equate to a contract of about $19.2 million, if calculated annually. He said financial terms of the extension — expected to go before the County Legislature’s Republican-controlled Rules Committee on Monday — match the “market rate.”

But Legis. Kevan Abrahams (D-Freeport), the minority party leader, accused Armor of “price-gouging” and called the proposal “an astronomical amount of money to pay for a provider that is inadequate.” He added: “Because the administration has handled this so poorly, it’s Armor or no one. We’re stuck basically.”

A state oversight agency found Armor provided inadequate care in connection with the deaths of at least five Nassau inmates since mid-2011. Four federal lawsuits against Armor and the county involving inmate fatalities are ongoing, after a jury’s nearly $8 million April verdict following a veteran’s 2012 jail suicide. The state attorney general sued Armor last year, before settling the claim that alleged deficient inmate care by fining it $350,000 and banning the company from New York contract bids for three years.

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As part of the contract change, the county wouldn’t charge Armor financial penalties, if it fails to meet performance standards. Ribando said the $250,000 Nassau got out of the state’s Armor settlement would cover any potential shortcomings.

Armor spokeswoman Teresa Estefan said in a statement Tuesday the company “is only considering an extension to aid the county in its transition to a new provider and ensure that there is continuity of patient care,” and called the idea that Armor is price-gouging “political grandstanding.”

Ribando said the amendment also makes the county 100 percent responsible for inmate hospitalization costs — a tab Nassau already covers up to $1 million before Armor picks up 60 percent of additional costs. But Ribando said those costs historically don’t surpass $1 million by August’s end.

The amendment also is intended to end litigation Armor brought against the county, seeking a judgment that it can leave May 31 and doesn’t have to stay to help transition to the next provider.

In the meantime, Ribando said the administration is “just working at a few minor details” in its negotiations with NUMC for a yearslong inmate medical care contract. Those talks began after no qualified vendors put in bids following a second request for contract proposals the administration issued in March.