President Trump said Wednesday he supports more background checks on gun purchases, as he traveled to the scenes of two deadly mass shootings in Dayton, Ohio and El Paso, Texas.

“There is a great appetite for background checks,” Mr. Trump told reporters at the White House. “I’m looking to do background checks. I think background checks are important.”

The president said “there is no political appetite” in Congress for a ban on so-called assault weapons of the type that were used in both shootings, which claimed 31 lives.

“You have to have a political appetite within Congress,” Mr. Trump said. “I have not seen it with regard to certain types of weapons.”

The first stop for the president and first lady Melania Trump was Miami Valley Hospital in Dayton, where they met with first responders, hospital workers, family members and some of the victims who were wounded in the attack on Sunday.

“You had God watching,” Mr. Trump said, according to White House press secretary Stephanie Grisham. “I want you to know we’re with you all the way.”

As Mr. Trump stepped off Air Force One in Dayton, Mayor Nan Whaley and Democratic Sen. Sherrod Brown of Ohio urged the president to call on Senate Majority Leader Mitch McConnell to bring the Senate back into session this week to vote on a House-passed bill that would expand background checks on firearms sales.

Mr. Brown said he also pressed the president for a public commitment to the legislation.

“I asked the president to promise to me, and to the American people, that he will sign that bill after he’s spoken out in support of it with Senator McConnell,” Mr. Brown told reporters. “He only said that ‘We will get things done.’”

The senator also asked Mr. Trump not to repeal Obamacare if he’s concerned about treatment of mental illnesses for people who potentially have access to guns.

At the hospital, Mr. Brown said, the president told a group of more than two dozen police and first responders that he wanted to give “honors and awards” to police who responded to the shooting.

Mr. Brown said he told the president in front of the group, “the most important thing you can do for these police officers is take these assault weapons off the streets.”

The mayor said she was glad that Mr. Trump didn’t visit the city’s Oregon entertainment district where the shooting took place.

“I think it was a good decision for him not to stop in the Oregon district,” she said. “A lot of people who own businesses in that district aren’t interested in him being there.”

On his way to Texas, the president disputed the accounts given by Mr. Brown and the mayor about his visit to Dayton. He said they were “totally misrepresenting what took place inside of the hospital.”

“Their news conference after I left for El Paso was a fraud,” Mr. Trump tweeted aboard Air Force One. “It bore no resemblance to what took place with those incredible people that I was so lucky to meet and spend time with. They were all amazing!”

In an address to the nation on Monday after the shootings, Mr. Trump didn’t mention universal background checks, focusing instead on “red flag” legislation that would keep guns out of the hands of people deemed high-risk or mentally ill.

Following last year’s mass shooting in Parkland, Florida, Mr. Trump edged toward endorsing background checks, calling them “common-sense measures that protect the rights of law-abiding Americans.” But after a meeting with officials from the National Rifle Association, he didn’t push for legislation.

As he departed the White House on Wednesday, the president didn’t specifically refer to a long-standing proposal by Sens. Patrick J. Toomey, Pennsylvania Republican, and Joe Manchin III, West Virginia Democrat, for expanded universal background checks. Mr. Trump did emphasize gun control in the context of preventing mentally ill people from owning firearms.

“I don’t want to put guns into the hands of mentally unstable people or people with rage or hate, sick people,” the president said. “I think we can bring up background checks like we’ve never had before. I think both Republicans and Democrats are getting close to a bill or doing something like background checks. I’m all in favor of it.”

He said he is in discussions with leaders in both parties, and it’s possible that Congress would interrupt its six-week-long August recess to address legislation.

“If we get close, I will bring them back,” the president said. “But … we have to see where we are with leadership. You have two sides that are very different on this issue. I’ve already got meetings scheduled. And I have had plenty of talks over the last two days. I think … we’re going to come up with something that’s going to be really very good, beyond anything that’s been done so far.”

White House senior counselor Kellyanne Conway said Wednesday that the Toomey-Manchin bill “could have bipartisan, bicameral support.”

The president said he doesn’t blame politicians such as Sen. Elizabeth Warren or Sen. Bernard Sanders who have been linked to supporters who carried out mass shootings. The shooter in Dayton reportedly was a supporter of Ms. Warren’s presidential candidacy and a left-wing devotee of the antifa extremist group.

“I don’t blame Elizabeth Warren, and I don’t blame Bernie Sanders, in the case of Ohio,” the president said. “I don’t blame anybody. These are sick people. These are people that are really mentally ill, mentally disturbed. It’s a mental problem.”

Democrats have blamed Mr. Trump in particular for the shooting in El Paso, where the gunman espoused white supremacist and anti-immigrant views.

“I don’t think my rhetoric does at all [incite violence],” the president said. “I think my rhetoric brings people together.”

Former Rep. Beto O’Rourke of Texas, a back-of-the-pack candidate for the Democratic presidential nomination, has been particularly harsh in his criticism of Mr. Trump’s anti-immigrant rhetoric, comparing it to the “Third Reich” and saying the president has “given people permission” to commit mass murder.

“My critics are political people, they’re trying to make points,” Mr. Trump said. “In many cases, they’re running for president and they’re very low in the polls. As much as possible, I’ve tried to stay out of that. I think we have toned it down. We’ve been getting hit left and right from everybody.”

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