The group said: "Devices designed to allow semi-automatic rifles to function like fully-automatic rifles should be subject to additional regulations."
Republicans have said they would consider banning the tool, despite years of resisting any gun control.
Lawmakers plan to hold hearings and consider a bill to outlaw the device.
The NRA called on Thursday for regulators to "immediately review whether these devices comply with federal law".
President Donald Trump later told reporters his administration would be looking into whether to ban them "in the next short period of time".
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"In the aftermath of the evil and senseless attack in Las Vegas, the American people are looking for answers as to how future tragedies can be prevented," NRA chiefs Wayne LaPierre and Chris Cox wrote in the statement.
They criticised politicians who are calling for gun control, writing that "banning guns from law-abiding Americans based on the criminal act of a madman will do nothing to prevent future attacks".
The statement, the organisation's first since Sunday's attack in Las Vegas that left 58 people dead and nearly 500 injured, noted that bump-stocks were approved by the Obama administration's Bureau of Alcohol Tobacco and Firearms.
The NRA's plan becomes clear
Analysis by Anthony Zurcher, BBC Washington
The NRA's strategy for responding to the Las Vegas mass-shooting is now coming into focus.
By recommending that an executive branch agency conduct a review of the legality of bump stock devices, the extremely influential gun rights lobby is seeking to direct efforts towards administrative, not legislative, solutions.
If Congress were to start drafting new laws, the process may be more difficult for the NRA to control. Democrats, who have been clamouring for the opportunity to debate new gun-control laws, could have their chance. Republican congressional leadership may try to clamp down on the proceedings, but there's a chance other proposals -like limits on magazine capacity, military-style rifle features and new background check requirements - could come up for consideration.
These types of provisions are popular with the public at large but vigorously opposed by the NRA and their supporters in Congress. It could make for difficult votes for some conservative legislators.
The White House and many congressional Republicans are pledging to have a "conversation" about the issue and "look into" the details. That, for the moment, is a far cry from action.
The NRA is now suggesting an alternate route.
White House spokeswoman Sarah Sanders, who spoke to reporters moments after the NRA statement was issued, said: "Members of both parties and multiple organisations are planning to take a look at bump-stocks. We welcome that and would like to be part of that conversation."
In the same statement the NRA urged Congress to pass their longstanding pet proposal to expand gun rights nationwide, so-called right-to-carry reciprocity.
The lobby group wants gun-owners with concealed-carry permits from one state to be allowed to take their weapons into any other US state, even if it has stricter firearms limits.
Another NRA policy priority, the deregulation of silencer attachments, appears to have stalled in Congress in the wake of the Las Vegas attack, after Republican sponsors withdrew their bill.
A bill to ban bump-stocks was submitted to the US Senate on Wednesday by California Democrat Dianne Feinstein.
A Republican-led version of the bill may be submitted for debate as early as Thursday, Florida Republican Carlos Curbelo told reporters.
He said there was growing bipartisan consensus and that his office had been "flooded" with calls from other lawmakers interested in the bill.
"I think we are on the verge of a breakthrough when it comes to sensible gun policy," he told reporters.
Bump-fire stocks, also called bump-stocks and slide-fire adapters, allow semi-automatic rifles to fire at a high rate, similar to a machine gun.
But they can be obtained without the extensive background checks required of automatic weapons.
Gun control 'a touchy subject'
By James Cook, BBC News, Las Vegas
Ask survivors of the Las Vegas massacre about gun control and you may well hear the sound of silence.
The cultures of country music and shooting overlap and many concert-goers remain strong supporters of the right to bear arms.
"It's obviously kind of a touchy subject," singer and performer Krystal Goddard, 35, told me after recounting the horror of her escape from the gig.
"I think that guns are just a symptom of other things going on," she said, although she added that she did not understand why anyone needed to own an assault rifle.
There is some support among survivors for banning bump-stocks but there is also a realisation that doing so does not amount to serious gun control.
And all the while the killing continues. Fifty-nine people died here on Sunday.
By Thursday afternoon at least 87 more people had been shot and killed across the US, according to the Gun Violence Archive. That's a Las Vegas massacre every three days.
Stephen Paddock, the gunman in Las Vegas, had fixed the accessories to 12 rifles used in his attack.
Bump-stocks typically cost less than $200 (£150) and allow nearly 100 high-velocity bullets to be fired in just seven seconds, according to one company advert.
One of the most popular manufacturers of bump-stocks, Slide Fire, said they had sold out "due to extreme high demands" since the Las Vegas shooting.
[BBC SOURCES]
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