In a tweet on Tuesday night, Mr Trump indicated he wants to strengthen the background check system, but offered no specifics.
Congress banned the sale and manufacture of machine guns for civilian use in 1986, and machine guns in circulation before then are tightly regulated, limited in number and expensive.
"We must move past cliches and exhausted debates and focus on evidence-based solutions and security measures that actually work", Mr Trump said. "We must do more to protect our children".
Bump stocks have only been a factor in one of the many high-profile mass shootings in recent years: last year's massacre in Las Vegas, now the deadliest in American history.
When asked if Mr Trump was open to reinstating a ban on assault-type weapons, White House spokeswoman Sarah Huckabee Sanders said officials "haven't closed the door on any front". The sentiment was echoed by the president on Tuesday in a post on his Twitter account.
"The president has expressed his support for the efforts to improve the federal background check system, and in the coming days, we will continue to explore ways to ensure the safety and security of our schools", added Sanders.
Read: 37-Page Indictment of Russians Interfering with US Elections
They are spending too much time trying to prove Russian collusion with the Trump campaign - there is no collusion . Attorney General Rod Rosenstein made all of those points abundantly clear in a press conference Friday.
Meanwhile, survivors of the Parkland high school shooting launched a campaign under the tag #NeverAgain, urging the U.S. government to ensure the saftey of the nation's schools. Trump will host students, parents and teachers involved in the Florida shooting and massacres at Sandy Hook Elementary School in CT and Columbine High School in Colorado for a listening session on Wednesday.
Facing rising political heat after the Parkland school massacre, Trump blamed his predecessors for a lack of action and said he would be different, despite widespread skepticism among gun control activists about his sincerity and capacity to make the case for change.
A Justice Department spokesman later said: "We look forward to the results of that process as soon as it is duly completed".
On Wednesday, Trump will host parents, teachers and students at the White House for a "listening session" that will include people impacted by mass shootings in Parkland, Columbine, Colorado and Newtown.
Bump stocks were attached to six of the long guns found in the Las Vegas shooter's hotel room.
"That is a gross infringement of Second Amendment rights", Gun Owners of America executive director Erich Pratt said in a statement.
Many Republicans and even the powerful conservative lobby of the National Rifle Association have supported some regulatory action on bump stocks.
Of course, the actual evidence says that Trump should go well beyond banning bump stocks if he's serious about preventing more deaths.