President Donald Trump is open to banning all guns for certain individuals following the shooting massacre at a Florida high school last week, a White House official told CNBC.

The White House would consider barring all firearms for some individuals, like those with mental illnesses, rather than guns like assault weapons for all Americans, according to the official. Many Democratic lawmakers have called for a ban on assault-style rifles and public support for such a policy has grown. However, the Trump administration is skeptical Congress would vote to block assault weapons.

As calls mount for action to tighten gun rules and protesters gather outside the White House, the Trump administration is "listening to everything," the official said. Trump will host students, parents and teachers involved in the Florida shooting and massacres at Sandy Hook Elementary School in Connecticut and Columbine High School in Colorado for a listening session on Wednesday.

Trump and many Republicans have previously argued mental illnesses drive mass shootings, rather than the guns themselves.

This week, the president has signaled his support for at least two narrow provisions to tighten gun laws. On Tuesday, Trump signed a memorandum recommending that Attorney General Jeff Sessions propose rules banning so-called bump stocks, which can effectively turn semi-automatic weapons into machine guns. The gunman who massacred 58 people at a concert in Las Vegas last year used such a device.

Trump also backed a bipartisan Senate bill to improve the existing background check system. It aims to make federal agencies better at following rules that require them to submit criminal convictions to the FBI, which could help stop high-risk individuals from getting guns.

"Whether we are Republican or Democrat, we must now focus on strengthening Background Checks!" Trump tweeted on Tuesday.

Lawmakers who back the bill say following existing procedures better may have stopped the shooter who killed 26 people at a church in Sutherland Springs, Texas, last year from getting a gun.

While the Senate may approve the legislation, it faces a tougher road to passage in the House. The chamber voted to improve background checks last year, but in tandem with another policy to allow people with concealed carry permits in one state to have a concealed weapon in any other state.

Concealed carry reciprocity likely would not get enough Democratic support to pass the Senate, while House conservatives may resist supporting a background check bill without it.

The influential National Rifle Association said it supported a ban on bump stocks last year following the Las Vegas mass shooting, the nation's deadliest in modern times. It also backed the bill to improve background checks, but in conjunction with the changes to concealed carry laws.

Many Democrats argue that both policies supported by Trump would not go nearly far enough to restrict access to guns.