US President Donald Trump supports efforts to improve background checks on gun ownership, the White House says.
He spoke with Republican Senator John Cornyn about a bipartisan bill that seeks to improve the checks in place before someone can buy a gun.
The suspect in last week's Florida school shooting, in which 17 people died, bought his gun legally. Nikolas Cruz appeared in court on Monday.
Students from the school have demanded action on gun control.
It has emerged that Mr Cruz was able to buy seven rifles in the past year, despite being examined by Florida mental health workers in 2016, US media report. One of those was the AR-15 semi-automatic rifle used to carry out last Wednesday's attack.
"While discussions are ongoing and revisions are being considered, the president is supportive of efforts to improve the federal background check system," White House press secretary Sarah Sanders said on Monday.
What checks are currently in place?
Federal licensed dealers must run checks on anyone trying to purchase a gun. A potential buyer fills in a form with personal information and criminal history.
That information is submitted to the FBI's National Instant Criminal Background Check System (NICS), which processed more than 25 million applications last year.
But the system has holes because it relies on state and federal officials to report any criminal convictions and mental health issues that should legally stop someone buying a gun to NICS.
Its failings were put in the spotlight last year after the US Air Force admitted it had failed to flag a gunman's domestic violence conviction before he killed 26 people at a church in Texas.
What changes are being proposed?
After that shooting, a bipartisan bill was introduced by Mr Cornyn and Democratic Senator Chris Murphy.
It would require federal agencies to report background information thoroughly and accurately. It also proposed offering states financial incentives to do the same through a penalty and reward system.
At the time, Mr Cornyn presented it as a fix to a "nationwide, systematic problem". But the bill has not been passed by Congress.
On Monday, Mr Murphy said the president's support for the bill was "another sign the politics of gun violence are shifting", but added that "no-one should pretend this bill alone is an adequate response to this epidemic".
What does Trump think about gun control?
Mr Trump's view on gun control has changed over time. He spoke out against gun control during the 2016 election race.
Mr Trump has repeatedly emphasised the role of the shooter's mental health in the Florida attack, but last year he repealed an Obama-era rule that allowed certain people who receive mental health-related benefits to be entered into a criminal database.
So many signs that the Florida shooter was mentally disturbed, even expelled from school for bad and erratic behavior. Neighbors and classmates knew he was a big problem. Must always report such instances to authorities, again and again!— Donald J. Trump (@realDonaldTrump) February 15, 2018
Mr Cruz, 19, a former student at Marjory Stoneman Douglas High School, had been investigated by the authorities for posting disturbing content online, according to reports.
What do his guardians say?
Mr Cruz has been living with the family of a high school friend since his adoptive mother died in November.
His guardians, James and Kimberly Snead, said on Monday that they had no idea that he had violent tendencies.
"I told him there'd be rules and he followed every rule to the T," Mr Snead, 48, told the South Florida Sun Sentinel newspaper.
"We had this monster living under our roof and we didn't know," his wife added.
The couple also said they knew Nikolas Cruz possessed weapons but they were not worried, as they had a gun safe and James Snead believed he had the only key.
What do the survivors from the attack want?
Students from the Marjory Stoneman Douglas High School have been vocal since Wednesday's attack about the need for changes gun laws.
One student, Emma Gonzalez, told the BBC that politicians were being handed a "blank slate" to act.
"Make something with it. We are giving you the chance to get on the right side of this."
Students have organised a march on Washington on 24 March, and want simultaneous demonstrations to happen nationally.
Other students across the US have also proposed other actions such as staging national school walk-outs. Dozens of high-school students staged a "lie-in" outside the White House on Monday and chanted "We want safe schools".
Mr Cruz appeared in court for the second time in Fort Lauderdale on Monday, charged with 17 counts of premeditated murder.
He sat with his head bowed during the short hearing, and was remanded in custody without bail.