WASHINGTON — Congressional Republicans, under intense pressure to respond to this weekend’s massacres, are coalescing around legislation to help law enforcement take guns from those who pose an imminent danger — a measure that, if signed into law, would be the most significant gun control legislation enacted in 20 years.

Such “red flag” laws might not be as momentous — or controversial — as the now-expired assault weapons ban or the instant background check system, both of which were enacted in 1994 as part of President Bill Clinton’s sprawling crime bill. The House, under Democratic control, passed far more ambitious bills in February that would require background checks for all gun purchasers, including those on the internet or at gun shows, and extend waiting limits for would-be gun buyers flagged by the instant check system.

[Democrats warn that federal “red flag” law may not be enough.]

But those bills have run into a blockade that Senator Mitch McConnell, the majority leader, has erected for House bills he opposes.

Now, after back-to-back shootings this weekend left 31 people dead in El Paso and Dayton, Ohio, Republicans who have long resisted gun restrictions appear rattled. In Ohio, Gov. Mike DeWine was shouted down on Sunday by mourners in Dayton demanding that he “do something.” On Tuesday, he urged his fellow Republicans in the state legislature to pass measures establishing red flag powers and expanding background checks.

“They were absolutely right,” Mr. DeWine said Tuesday morning at a news conference. “We must do something, and that is exactly what we are going to do.”

Representative Michael R. Turner, a Republican whose district includes Dayton, went further.

“I will support legislation that prevents the sale of military-style weapons to civilians, a magazine limit and red flag legislation,” he said in a statement Tuesday. “The carnage these military-style weapons are able to produce when available to the wrong people is intolerable.”