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 Mr. Trump, are embracing the concept, and Senator Lindsey Graham, Republican of South Carolina, is drafting a bill to develop a federal grant program to help states pass and implement 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. “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 and El Paso, told reporters he was open to expanding background checks for gun purchases. But Senator Mitch McConnell, the majority leader, has blocked consideration of the House bills.

The gunman’s motive in Dayton may not be tied to politics, the authorities say.

Although the F.B.I. is investigating “violent ideologies” that the Dayton gunman was exploring, the authorities have said there was no evidence that his rampage had anything to do with his political views, which his friends have described as far-left.

The gunman, Connor Betts, who was killed by the police, espoused leftist views online and in conversations with friends, promoting socialism and the idea that liberals should own guns. And a Twitter account that is believed to be his but has not been confirmed by the authorities showed support for antifa, the loose group of people who call themselves “anti-fascists” and often believe that violence against people they view as “fascists” is justified.

In the wake of the mass shooting that killed nine and wounded more than two dozen others, conservatives have pointed out that the Twitter account that may be associated with the gunman also expressed support for Mr. Sanders and Ms. Warren, sharing one post that said they were the only two acceptable candidates in the race for president.

This week, a federal law enforcement official said the F.B.I. was looking at whether the gunman was associated with incel, or involuntary celibate, groups, which are generally made up of misogynists who disparage women online, in part for refusing to have sex with them.