The Cuomo administration has proposed extending out-of-network coverage requirements for emergencies and specialists to all health insurers in New York in what it says is an effort to protect consumers from big surprise medical bills.

The requirements currently apply to health maintenance organizations and 16 insurance plans in New York’s new health exchange, according to the state Department of Financial Services. When a network lacks an available specialist, or in emergencies, patients could get treated elsewhere and pay only their usual insurance fees.

Legislation that Gov. Andrew Cuomo recently proposed as part of a budget for the fiscal year starting April 1 would apply those consumer protections to non-HMOs, which provide most health coverage in New York.

If approved by the Legislature, it would also require advance disclosures by hospitals and doctors to patients about who’s actually in their insurance networks. Payment disputes would go to arbitration between the providers and insurance companies, leaving the patients out.

The protections wouldn’t apply to consumers who simply opt for providers outside their networks.

The New York Health Plan Association, an insurers’ group, and the Medical Society of the State of New York, which represents doctors, have expressed respective concerns about high doctor billing and low coverage amounts.

Concert pianist Claudia Knafo, whose four-hour disc surgery two years ago left her with $97,000 in bills and dunning notices, visited the Capitol on Thursday to advocate passage.

“I was concerned that to get through this I would have to declare bankruptcy,” she recalled.

According to Knafo, the surgeon said he was in her insurance plan but wasn’t. Her insurance initially paid almost $67,000, then demanded back all but $3,500. She tried for months to resolve it and hired a lawyer. The company finally dropped its claim after she called DFS Superintendent Ben Lawsky.

She declined Thursday to name the doctor, hospital that misidentified the doctor as a member of her insurance plan, or the insurer.

Lawsky’s department issued a report two years ago examining unexpected medical bills, noting that it received 2,105 reimbursement complaints in 2011 and that insurers reported another 1,310 that year.

Advocates for the legislation said there are certainly many more patients who’ve had similar problems and didn’t know where to complain. “These are just the tip of the iceberg,” said Chuck Bell, program director for Consumers Union.

Nationally, about one in three consumers pays extra for out-of-network insurance coverage, Bell said. The proposed law would require insurers offering that in New York provide at least one option covering 70 percent co-insurance of usual, customary and reasonable fees.

Blair Horner, of the New York Public Interest Research Group, said corrective legislation has been stalled so far by two big powerful interest groups, the insurers and the medical providers.

Elizabeth Dears, senior vice president of the Medical Society, said the group has sought a threshold of 80 percent of usual and customary costs, instead of letting insurers decide when a bill is reasonable. “If we need a dispute resolution, it needs to focus on outlier bills,” she said.

Leslie Moran, senior vice president of the New York Health Plan Association, said the group doesn’t want a threshold and instead wants to let the insurance plan and provider negotiate, with an independent arbitrator settling disputes by choosing one proposal or the other. It also wants options in reimbursement methods and to see doctors required to tell patients their prices upfront, not just when asked.

The consumer advocates said the issues between those groups involve arbitration terms and payment rates, including possible mandatory minimums. They said Cuomo’s proposed legislation improves its prospects this year.

“I would argue this is one of the issues you should hold the budget up over,” Horner said. “Real people lose their houses when these things happen.”