WASHINGTON — Congressional Republicans moved on Tuesday to defuse President Trump’s threat to cut off critical payments to health insurancecompanies, maneuvering around the president toward bipartisan legislation to shore up insurance markets under the Affordable Care Act.

Senator Lamar Alexander of Tennessee, the influential chairman of the Senate Health, Education, Labor and Pensions Committee, announced that his panelwould begin work in early September on legislation to “stabilize and strengthen the individual health insurance market” for 2018. He publicly urged Mr. Trump to continue making payments to health insurance companies to reimburse them for reducing the out-of-pocket medical expenses of low-income people.

In the House, two Republicans, Representatives Tom Reed of New York and Charlie Dent of Pennsylvania, teamed with Democrats to promote incremental health legislation that would also fund the cost-sharing subsidies.

The moves were a remarkable response to the president’s repeated threats to send health insurance markets into a tailspin. They offered tangible indications of cooperation between the parties after Republican efforts to scrap the Affordable Care Act collapsed in the Senate last week, all but ending the seven-year Republican quest to overturn President Barack Obama’s signature domestic achievement. Lawmakers from both parties concede that the health law needs improvement, as consumers face sharp premium increases and a shrinking number of insurance options in many states.

These problems have been exacerbated by a president who has publicly predicted that the Affordable Care Act will “implode” and appears determined to help fulfill that prophecy. Mr. Trump has repeatedly threatened to cut off the subsidies, known as cost-sharing reduction payments, which reimburse insurers for cutting deductibles and other out-of-pocket costs for millions of low-income people. Without them, insurers would almost certainly raise premiums not only for poor consumers but also for many other people buying plans on the individual insurance market.

In California, the state agency that runs the insurance marketplace announced on Tuesday that rates would increase by 12.5 percent on average next year. That is slightly lower than the rate increases Californians saw this year. But Peter V. Lee, the executive director of the agency, Covered California, said the average increase would be twice as high for popular “silver” plans if the Trump administration blocked the cost-sharing payments.

“This policy allowed health plans to stay in the market when they might have left otherwise,” Mr. Lee said of the potential additional increase, which he called a “surcharge.”

He added, “By the end of this month, we have to hear there’s clarity that the cost-sharing reductions will be made, or we will apply the surcharge.”

In Kentucky, according to data posted by the federal government, Anthem has requested rate increases averaging 34 percent for plans covering 69,500 people. BlueCross BlueShield of South Carolina has sought rate increases averaging 33 percent.

And Blue Cross and Blue Shield of Texas is seeking rate increases that average about 23 percent, and it said 389,800 people may be affected. The company cited uncertainty about cost-sharing subsidies as a factor, along with medical inflation.

In the House, a group of members known as the Problem Solvers Caucus announced agreement this week on a bipartisan set of proposals to stabilize insurance markets and revise the Affordable Care Act to provide relief to consumers and small and midsize businesses. The proposals would provide money for cost-sharing reduction payments, repeal a tax on medical devices and exempt businesses with fewer than 500 employees from the law’s requirement to offer health insurance to workers.

“My hope is that maybe the president will take this into consideration in regards to the upcoming decision” on cost-sharing subsidies, said Mr. Reed, a co-chairman of the caucus, whose members are split roughly evenly between the two parties.

“Many in our parties don’t want us to do this,” Mr. Reed said of the bipartisan initiative. “Many of us still retain our philosophical opposition and substantive opposition to the Affordable Care Act. But it’s clear to us that what we have to do is come together, find that common ground and govern for the American people.”

Under the proposal, funds for the cost-sharing payments would be guaranteed, and Congress could review use of the money each year, just as it reviews other federal spending.

Mr. Alexander said that it was important for Mr. Trump to approve the payments for August and September, and that Congress should approve “in a bipartisan way” a continuation of the payments for at least a year.

“Without payment of these cost-sharing reductions,” he said, “Americans will be hurt. Up to half of the states will likely have bare counties with zero insurance providers offering insurance on the exchanges, and insurance premiums will increase by roughly 20 percent, according to America’s Health Insurance Plans,” a trade group for insurers.

Senator Patty Murray of Washington, the senior Democrat on the health committee, welcomed Mr. Alexander’s statement.

Mr. Alexander said the committee would hold hearings starting the week of Sept. 4 “on the actions Congress should take to stabilize and strengthen the individual health insurance market, so that Americans will be able to buy insurance at affordable prices in the year 2018.”

He said the committee expected to hear from state insurance commissioners, patients, governors, health care experts and insurance companies.

Mr. Alexander said his proposal was a necessary response to an imminent crisis.

“In my opinion,” Mr. Alexander said, “any solution that Congress passes for a 2018 stabilization package would need to be small, bipartisan and balanced. It should include funding for the cost-sharing reductions, but it also should include greater flexibility for states in approving health insurance policies.”

Payment of the cost-sharing subsidies is a top priority for insurers and for Democrats in Congress, who say that cutting off the payments would cause havoc in insurance markets.

The president has the power to stop the payments because a federal judge ruled last year that the Obama administration had been illegally making the payments, in the absence of a law explicitly providing money for the purpose.

The Obama administration appealed the ruling, and the case is pending before the United States Court of Appeals for the District of Columbia Circuit.

House Republicans, who filed suit to stop the payments in 2014, and the Trump administration have told the court that they are discussing “measures that would obviate the need for judicial determination of this appeal, including potential legislative action.”

The appeals court on Tuesday allowed California, New York and 15 other states to intervene in the case. The states have shown a substantial risk that termination of the cost-sharing payments “would lead directly and imminently to an increase in insurance prices, which in turn will increase the number of uninsured individuals for whom the states will have to provide health care,” the court said.

Mr. Alexander said he hoped Congress would eventually approve long-term measures to create a more robust market for people who buy insurance on their own. But first, he said, “we need to put out the fire in these collapsing markets.”

The cost-sharing payments help people with incomes of 100 percent to 250 percent of the federal poverty level (about $12,060 to $30,150 a year for an individual). But some Republicans say that providing the money would amount to “a bailout for insurance companies,” in the words of Senator Ted Cruz of Texas.

“It’s what the Democrats want,” Mr. Cruz said on Tuesday. “The Democrats are the party of the big insurance companies.”