WASHINGTON — Congressional Republicans moved on Tuesday to defuse President Trump’s threat to cut off critical payments to health insurance companies, 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 panel would 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.