In the House, a group of Republicans on the Judiciary Committee — led by Representative Jim Jordan of Ohio — was looking at its own plans to reimagine police training, increase accountability for officers who use improper force or violate the rights of civilians, and collect new data to track the behavior by departments across the country.

But privately, Republican lawmakers and aides conceded they had few proposals ready to offer and were instead racing to reach a consensus about how to proceed. Late Tuesday afternoon, Mark Meadows, the White House chief of staff, rushed to Capitol Hill with Jared Kushner, Mr. Trump’s son-in-law and senior adviser, to huddle with Republicans about what they could quickly agree on.

Mr. Meadows told reporters after that meeting that Mr. Trump was in favor of overhauling policing laws “sooner rather than later.” But he declined to name any specific legislative priorities the White House would support, saying the administration would be “responsive” to suggestions made by “stakeholders” on Capitol Hill.