WASHINGTON — Senator Chuck Schumer, the Democratic leader, put the brakes on Republicans’ quick embrace of “red flag” laws as a response to last weekend’s gun violence, saying on Wednesday that any gun-related legislation moving through the Senate must be accompanied by a House bill requiring background checks on all gun purchasers.

Red flag laws allow the authorities to obtain a special type of protective order — known as an extreme risk protection order, or E.R.P.O. — to remove guns from people deemed dangerous. Republicans, including President Trump, are coalescing around the concept, and Senator Lindsey Graham of South Carolina, the chairman of the Senate Judiciary Committee, is drafting a bill to develop a federal grant program to help states pass and carry out such laws.

“We Democrats are not going to settle for half measures so Republicans can feel better and try to push the issue of gun violence off to the side,” Mr. Schumer said in a statement. “Democrats in the Senate will seek to require that any E.R.P.O. bill that comes to the floor is accompanied by a vote on the House-passed universal background checks legislation.”

Mr. Trump, on his way to Dayton, Ohio, and El Paso, where back-to-back massacres over the weekend left 31 people dead, told reporters he was open to expanding background checks for gun purchases — a surprising development, given that the National Rifle Association, the nation’s largest gun lobby, and many Republicans are vehemently opposed.