To Democrats, the fact that the N.R.A. is not opposed to the proposal is prima facie evidence that it falls short. They are demanding a more robust debate over a series of gun initiatives, notably what they call a “universal” background check system that would cover all gun transactions in the country. That plan is opposed by the gun lobby but was seemingly endorsed by Mr. Trump, putting him at odds not only with the gun advocacy group that strongly backed him, but with many congressional Republicans, as well.

Senator Christopher S. Murphy, Democrat of Connecticut and an author, with Mr. Cornyn, of the more limited background check bill, equated taking up his own legislation without allowing consideration of alternatives to “slamming the door in the face of all these kids who are demanding change.”

“I think it is imperative that we rise to the moment,” Mr. Murphy said.

Mr. Trump told lawmakers they should use a broader bipartisan background check measure that failed in 2013 after the school shooting in Newtown, Conn., as the basis for a comprehensive measure that could take in many proposals, including the Cornyn plan, new mental health provisions and added security for schools. He urged backers of a renewed ban on assault weapons to make their case to the authors of the legislation and said he would welcome the opportunity to enact one sweeping bill to combat mass shootings. But Congress has for years been unable to make even incremental headway on new gun buying limits, let alone the type of multifaceted plan called for by the president.

Democrats would like to persuade Senator Mitch McConnell, Republican of Kentucky and the majority leader, to allow votes on major gun control proposals, a plan that could throw the Senate into a full-blown showdown over competing initiatives. But Mr. McConnell is never one to rush into a fight that divides his own party if he can help it. While Mr. McConnell has been quiet about his intentions, Mr. Cornyn opened the door to that possibility of a wider debate provided it could get to passage of his bill at a minimum.

“We can set up a situation where they can vote on those amendments,” he said. “What I don’t want to do is leave here this week and go back home to Texas and say we failed to do anything to try to address these tragedies.”

As lawmakers quibbled about how to proceed, some of the multiple proposals promoted by Mr. Trump were falling by the wayside.