GOP Sen. Patrick Toomey of Pennsylvania and Democrats Joe Manchin of West Virginia and Chris Murphy of Connecticut cautioned that they did not win Trump's endorsement of their background check bill during their 40-minute telephone conversation. But they nonetheless depicted the president as engaged and encouraging, with Toomey saying Trump has a strong "interest in doing something meaningful" to prevent gun violence, even though he has not endorsed any specific approach.

Trump, speaking to reporters in the Oval Office, said he and lawmakers were "working very, very hard together" and "seeing if we can come up with something that's acceptable to everybody."

But he offered few clues about what he is willing to put his weight behind, even as he insisted that "progress is being made. "

"We're looking at background checks and we're looking at putting everything together in a unified way so that we can have something that's meaningful," he said, adding that, "At the same time, all of us want to protect our great Second Amendment. It's very important to all of us."

Asked specifically about the bill in question, which would require background checks for firearm purchases at gun shows and over the internet, Trump said there are "a lot of things under discussion," including "some things" that "will never happen" and other things that might.

"It's really 'Gun Sense,' if you think about it," Trump said, adding: "We're having great dialogue. We'll see what happens."

The bill in question failed to pass in 2013 and many Republicans continue to oppose the idea as an infringement on gun rights. The senators said they hoped to hear back from the White House Thursday.

Over in the House, No. 2 Democrat Steny Hoyer of Maryland said that the chamber would vote in October on a "red flag" bill aimed at keeping guns out of the hands of mentally ill people or those convicted of hate crimes, along with legislation to ban high capacity gun magazines.

Hoyer, though, was noncommittal about action on an assault weapons ban that is backed by many liberals in his party.