“The House has not made much progress toward actually making a law, just more resistance theater,” Senator Mitch McConnell of Kentucky, the majority leader, said on the Senate floor Wednesday morning. “The Senate has a better and more bipartisan way forward.”

“It’s a productive compromise that would go a long way to begin to address the border crisis,” Mr. McConnell added. “No poison pills, just a clean bill.”

Mr. Trump voiced his displeasure with the House bill on Wednesday morning, saying on Fox Business Network that he was “not happy with it because there is no money for protection.”

To make their point, Republican Senate leaders put the House’s $4.5 billion bill to a test vote; it failed, 37 to 55, with three Democrats voting against the measure. Seven Democrats, all presidential candidates, were not present ahead of the first Democratic debate in Miami on Wednesday night.

But Senator Chuck Schumer of New York, the Democratic leader, suggested a few changes to the Senate bill could win support among House Democratic leaders in time for quick final passage.

The Senate legislation would allocate about $1.3 billion to improve facilities at the border and $2.9 billion for the care of migrant children. The measure prohibits the Department of Homeland Security from adding more beds at detention centers or migrant processing facilities, ostensibly to slow the immigration crackdown. The Senate would require the department to allow congressional visits to facilities housing unaccompanied children with two days’ notice; the House bill would permit them with no notice at all.