—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.

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.”But in the Senate, where a background checks bill failed in 2013 after 26 children and staff members were gunned down at Sandy Hook Elementary School in Newtown, Conn., red flag laws may be the only gun-related measure that could squeeze through.

President Trump endorsed the idea on Monday in a speech from the White House, giving skittish Republicans cover to embrace it.The laws authorize courts to issue orders allowing police to temporarily confiscate firearms from a person deemed by a judge as posing a risk of violence.

Often, requests for the orders come from relatives and friends concerned about a gun owner who expresses suicidal thoughts or threatens to harm others.

Senator John Thune of South Dakota, the No. 2 Republican, told his hometown newspaper, The Argus Leader, that he was “confident Congress will be able to find common ground on the so-called red flag issue.”

Senator Lindsey Graham, Republican of South Carolina, has already proposed legislation that would offer federal grants to states to help them enact and enforce red flag laws, also known as “extreme risk protection orders

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