By Ashley Killough and Deirdre Walsh, CNN

(CNN) -- The US House of Representatives will vote Wednesday on a bill to fund more security at schools, exactly one month after a gunman killed 17 people at a high school in Parkland, Florida.

While the bill is expected to pass, many Democrats are frustrated that it doesn't include any gun control measures amid stepped up public debate over how to combat gun violence.

"This is a pretense that we are doing something while assuring the NRA that we aren't doing anything," Rep. Steny Hoyer of Maryland, the second ranking Democrat in the House, told reporters Tuesday.

On the Senate side, the Judiciary Committee will hold a widely anticipated hearing Wednesday on the Florida shooting and failures by the FBI and law enforcement to act on warning signs displayed by the gunman before the shooting.

Both of Florida's senators -- Republican Marco Rubio and Democrat Bill Nelson -- will testify as witnesses at the hearing. The two men have also teamed up on legislation that would encourage states to adopt so-called red flag laws, which would give law enforcement the authority to seize guns from people who pose a threat to themselves or others.

It's one of many gun control bills proposed by members on both sides of the aisle, but most efforts have largely stalled.

President Donald Trump reiterated his support last weekend for a bill by Sen. John Cornyn of Texas, the No. 2 Republican in the Senate, and Democratic Sen. Chris Murphy of Connecticut that would encourage states and federal agencies to enter more data into the National Instant Criminal Background Check System, known as "Fix NICS."

It was first introduced last fall after the Sutherland Springs, Texas, church shooting but it has seen renewed attention since last month's Florida shooting. While the bill currently has more than 60 cosponsors -- a normal indicator that it could avoid a filibuster -- many Democrats want to open up the legislation to amendments, and it's unclear how Republican leaders will proceed.

"I'm extremely interested in seeing Senator Cornyn's Fix NICS bill passed and a significant school safety bill passed," House Majority Leader Mitch McConnell told reporters on Tuesday. "The best way to get that done is still under discussion. But I'm anxious to pass both of them, and pass both of them soon."

The witness list at the Senate Judiciary Committee hearing will also include David L. Bowdich, the acting deputy director of the FBI, as well as Ryan Petty, whose daughter was killed in the Florida shooting, and Katherine Posada, a teacher at the school, Marjory Stoneman Douglas High School.

And while there are no gun industry witnesses at the first hearing since the shooting, they loom large in the debate and play a critical role in lobbying members of Congress and rallying their supporters across the country during elections.

The House bill, the STOP School Violence Act, would provide more training for school officials and local law enforcement to respond to mental health crises, as well as, among other things, money to develop anonymous reporting systems for threats and deterrents like metal detectors and locks.

It does not include many of the components of a proposal unveiled by the White House -- most notably it does not include any provisions to arm teachers. House Republicans have largely ignored the President's plan, especially since he publicly declared that the major legislation the GOP-controlled chamber passed in December to loosen concealed carry rules was not something that could pass as part of broader gun legislation.

"This is about schools but it's not just about schools," Rubio told reporters Tuesday at a news conference about the Senate version of the bill. "When someone is determined that they're going to commit an act of violence, it could be in a school, it could be in a mall, it could be in a movie theater, it could be in an airport, it could be at a stadium. So what we're really focused on here more than anything else is identifying the people that are going to commit a violent act irrespective of where they're going to commit it and stopping them before they do it."