President Trump said Sunday he will announce steps to stop gun violence, as top Democrats raced to blame him for fomenting two mass shootings this weekend in Texas and Ohio with a toxic stew of violence and white nationalism, and pressed him for quick action on gun control.

After conferring Sunday with Attorney General William P. Barr, FBI Director Christopher A. Wray, state officials and lawmakers, the president said he will outline his proposals around 10 a.m. Monday.

“We’re going to take care of it,” Mr. Trump told reporters in New Jersey before returning to Washington. “A lot of things are in the works. I’ve spoken to members of Congress about whatever we can do, and a lot of things are being done right now as we speak.”

He suggested that part of his approach will focus on mental health laws and gun ownership.

“This is also a mental illness problem,” Mr. Trump said. “If you look at both of these cases, this is mental illness. These are really people that are very, very seriously mentally ill.”

While Republicans struggled to present a unified response to the violence, several Democratic presidential candidates accused Mr. Trump of inciting white supremacists with his words and policies, including attempts to impose stronger security on the border with Mexico and his engagement in feuds with black and Muslim Democratic lawmakers.

“This president is encouraging greater racism, and not just the racist rhetoric but the violence that so often follows,” former Rep. Beto O’Rourke of Texas, a 2020 Democratic presidential candidate, said on CNN. “Calling Mexican immigrants rapists and criminals, warning of an invasion at our border, seeking to ban all people of one religion. It encourages the kind of violence that we’re seeing, including in my hometown of El Paso.”

Sen. Bernard Sanders of Vermont said Mr. Trump “has to stop that racism and that xenophobia immediately.”

“I am sure that President Trump does not want anybody in this country to go around shooting other people,” Mr. Sanders told Jake Tapper on CNN’s “State of the Union.” “But what he has got to understand is that when you have language that is racist, that is virulently anti-immigrant, there are mentally unstable people in this country who see that as a sign to do terrible, terrible things.”

Addressing Democrats’ accusations, the president said, “Hate has no place in our country. These are two incredible places. We love the people.”

The president also said in his brief remarks that the problem of mass shootings in the U.S. predates his administration.

“We have to get it stopped,” Mr. Trump said. “This has been going on for years.”

Acting White House Chief of Staff Mick Mulvaney said the administration is willing to discuss “the role of guns” but added, “To think that this is just a gun issue that many people make it out to be is not right.

“We’ve had guns in this country for hundreds of years. We haven’t had this until recently, and we need to figure out why,” Mr. Mulvaney said on ABC’s “This Week.” “If we can agree on one thing as a nation — Democrat, Republican and independent, I don’t care — it is that crazy people like this should not have been able to get guns.”

Mr. Trump, who spent the weekend at his golf club in Bedminster, New Jersey, ordered U.S. flags to be flown at half-staff as a sign of respect for the victims of the shootings, which killed at least 29 people and wounded dozens of others.

The shootings also could become an international incident.

Mexican Foreign Minister Marcelo Ebrard said his country would take “forceful legal actions” to “demand that conditions are established that protect … Mexicans in the United States.”

Mr. Ebrard called the attack in El Paso, Texas, on Saturday “a terrorist act” against Mexicans — international law requires governments to protect foreign citizens against native vigilantes and pogrom-style attacks — and urged the U.S. government to “set a clear position against hate crimes.”

He said his country also plans legal action against the manufacturer of the weapon used in Saturday’s attack and urged the U.S. to tighten its gun control laws.

Mexican officials said six of the 20 people killed and at least nine of the 26 people wounded at a Walmart in the Cielo Vista Mall in El Paso, Texas, were Mexican nationals. Even apart from matters related to legal and illegal immigration, El Paso lies directly across the Rio Grande from Ciudad Juarez, and the two cities combine to create a metropolitan area of more than 2 million people where cross-border traffic and commerce are part of the routine.

El Paso Police Chief Greg Allen said authorities think the suspect, 21-year-old Patrick Crusius, posted a screed against immigration titled “The Inconvenient Truth” about 20 minutes before the shooting started. The essay called the massacre a response to an “invasion” of Hispanics and a way to “fight to reclaim my country from destruction.”

The gunman in the Dayton, Ohio, attack Sunday, 24-year-old Connor Betts, did not seem to have any obvious political motive. He opened fire in an entertainment district around 1 a.m. and killed his sister and eight other people. Within 30 seconds, he was fatally shot by police.

Many U.S. lawmakers urged Senate Majority Leader Mitch McConnell, Kentucky Republican, to cancel the Senate’s August recess to take up two House-passed gun control measures.

“This is a time that demands not words but actions,” said Rep. David N. Cicilline of Rhode Island, a House Democratic leader who also called for a ban on assault weapons. “As we have seen time and time again, it is far too easy for bad people to buy guns in our country. Congress must act now.”

Mr. McConnell, who wasn’t expected to call the Senate back, suffered a broken shoulder at home Sunday when he tripped on his patio, his spokesman said.

“He has been treated, released and is working from home in Louisville,” said communications director David Popp.

Mr. McConnell contacted Sen. John Cornyn, Texas Republican, and Sen. Rob Portman, Ohio Republican, “to express his deepest sympathies for the people of El Paso and Dayton and discuss the senseless tragedies of this weekend,” Mr. Popp said.

“The leader will continue to work from home,” he added.

Many top Democrats were unified in directing their criticism at Mr. Trump after the party’s presidential candidates spent much of last week attacking one another in their second primary debates.

Sen. Cory A. Booker of New Jersey lamented a lack of will to name and confront “right wing” and “white supremacist” violence in the U.S.

“Donald Trump is responsible for this,” Mr. Booker said on CNN. “He is stoking fears and hatred and bigotry. He is failing to condemn white supremacy.”

Mayor Pete Buttigieg of South Bend, Indiana, said the kind of hatred that produced the two shootings “is being legitimized from on high.”

“This is part of a climate where people who are in the grip of this hateful extremist ideology feel validated, and they feel validated from all the way at the top,” he said on “Fox News Sunday.”

House Judiciary Committee Chairman Jerrold Nadler of New York said he was “anguished” that the online essay before the El Paso attack included rants about a “Hispanic invasion.”

“This manifesto appears to echo the same anti-immigrant sentiment that permeates the language espoused by President Trump,” Mr. Nadler said. “Ultimately, it is the perpetrators who bear responsibility for their heinous acts of domestic terrorism, but President Trump’s racist rhetoric has stoked the flames of hate and white supremacy, and increased the likelihood that people will commit mass murder based on these evil notions. The President must stop his racist rhetoric that has the effect of encouraging mass murder.”

Few Republican lawmakers came forward with proposals.

Mr. Portman said at a press conference in Dayton that Congress should “learn lessons” from the shootings.

“Are there more things that could be done? I’m sure there are,” Mr. Portman said. “But I will say there’s something deeper going on here.”

He mentioned high suicide and addiction rates in the region and the need for better mental health services. He also said the community has been “ground zero in terms of the opioid crisis.”

In the Dayton shooting, the city’s police chief said nothing in the shooter’s background would have precluded him from purchasing a firearm legally.

House Minority Leader Kevin McCarthy, California Republican, said his party has taken action on gun violence. He noted that Mr. Trump signed into law in March 2018 a measure that calls for penalties to government agencies for not reporting to the National Instant Criminal Background Check System and that the president outlawed rapid-fire “bump stock” accessories after the Las Vegas mass shooting in October 2017.

Said Mr. Trump, “We have done much more than most administrations … but perhaps more has to be done.”

Two former House Republican lawmakers, Trey Gowdy of South Carolina and former Ohio Gov. John Kasich, spoke in favor Sunday of expanded background checks for firearms purchases. One measure passed by the House this year would extend the review period for background checks on gun purchases from three days to 10 days.

After a mass shooting at a high school killed 17 in Parkland, Florida, in February 2018, Mr. Trump convened a federal commission on school safety. The panel recommended that schools consider arming themselves and take more action to address students’ mental health issues.

The president said the quick efforts of law enforcement officers in El Paso and in Dayton prevented the death tolls from being even higher.

“As bad as it was, it could have been so much worse,” he said. “The job they’ve done is incredible.”

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