House Democrats on Tuesday started a push for new gun control measures in the wake of several mass shootings during their summer recess. Advocates are making their voices heard

Joseph Sakran spoke from experience Tuesday. Shot in the neck at a high school football game, he was inspired to become a trauma surgeon. "The worst part of my job is having to walk into those waiting rooms and look at those parents and tell them that their child is never coming home," Sakran said.

Doctors and gun victims and mayors are blanketing the Capitol as some Republicans signal new openness to gun laws after a summer of deadly mass shootings. Republican leaders met with the president Tuesday to try to nail down what he supports.

"I'm hoping in the next few days, early next week we'll have a breakthrough," Senator Lindsey Graham said.

"He's very interested," Senator Pat Toomey said. "I think he's learning about this issue."

House Democrats aren't waiting by advancing bills that would ban high-capacity magazines and confiscate guns from high-risk individuals, they would also urge the Senate to pass universal background checks.

"We won't stop until the job is done," Senate Majority Leader Nancy Pelosi said.

House Gun Violence Prevention Task Force
The Reverend Sharon Risher, who lost her mother and cousins in the 2015 church shooting in Charleston, wipes tear on Tuesday, September 10, 2019.  GETTY

Finding consensus won't be easy. CBS News asked Senate Majority Leader Mitch McConnell if he thought it would be reasonable to mandate background checks for all commercial gun sales.

"What I've said is that we are waiting to see how we can actually achieve something on this issue," he said. "I'm going to wait and assess the proposal that actually could become law."

McConnell said there's no point endorsing any particular policy until he knows what the president will sign. But Democrats said they won't believe the GOP is serious bout this issue until it starts getting specific about what it's willing to do.