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.