HOUSTON — At a sprawling convention center-turned-shelter where evacuees from Hurricane Harvey have sought refuge, lawmakers pledged on Monday that Congress would begin voting on disaster relief legislation by midweek.

“The first order of business is to make sure that we do everything we can to make sure the federal government is providing the support that’s needed for Texas and Texans to heal,” said Senator John Cornyn, Republican of Texas, after he and other members of Congress met with residents who were forced from their homes by the storm.

A show of bipartisan urgency, with the House scheduled to start voting Wednesday morning on an initial round of relief money, comes as Texans begin to shift from emergency response to long-term recovery. Officials from both parties are seeking to assure storm victims that the federal government will stay to help. But many questions loom about the size and scope of federal aid.