WASHINGTON — The House moved toward approving legislation on Wednesday to avert a nationwide rail strike by imposing a labor agreement between rail companies and their workers, as lawmakers rushed to shield the economy from the threat of a holiday-season work stoppage and prevent a disruption in shipping across the country.

Acting quickly the day after President Biden made a personal appeal at the White House, the House opened debate on a bill that would force the rail companies and employees to abide by a tentative agreement that the Biden administration helped broker earlier this year, which increased pay and set more flexible schedules for workers. But with liberal Democrats threatening to withhold their votes unless the legislation granted paid leave, a key demand of workers that was omitted from that agreement, lawmakers were also considering a separate measure to add seven days of compensated sick time to the compact.

Multiple unions have balked at the tentative agreement because of the lack of paid family or medical leave, which workers have insisted is vital to help counter the toll of unpredictable schedules.

If it passes the measure on Wednesday as expected, the House would take the first step to compel all 12 unions to accept the deal.

Leaders in both parties said they were reluctant to intervene in a labor dispute, and some Democrats — particularly progressives — were deeply frustrated about being called upon to override the will of rail workers pressing for basic workplace rights. But the threat of economic damage, as well as Mr. Biden’s personal appeal for Congress to act, appeared to have provided the momentum necessary to propel the measure with unusual speed.

“After hearing from our members, we are in agreement that a nationwide rail strike must be prevented — and that more must be done to secure the paid sick leave that hard-working railroaders deserve,” Speaker Nancy Pelosi of California wrote in a letter to members of the House late Tuesday. She called for a “a strong, bipartisan vote to support our legislation — which will give America’s families and businesses confidence in our economy this holiday season.”

Both measures were expected to pass the House on Wednesday and go to the Senate, where leaders in both parties have indicated they would move quickly to avoid a disruption to the nation’s rail service.

It was unclear whether the paid leave proposal had the bipartisan support necessary to pass the Senate, but by approving it, the House left open the possibility that it could ultimately be added before the tentative agreement was cleared for Mr. Biden’s signature.

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