"I think we are on the verge of a breakthrough when it comes to sensible gun policy," Curbelo told reporters Thursday, saying his office has been "flooded" with calls from other lawmakers asking about the bill.
He said that the shooting in Las Vegas, where the shooter appeared to modify several weapons with bump stocks to fire rounds more quickly, demonstrated that these devices, are in his opinion at the "outer limits of what the law allows, in this case something that we strongly believe should be illegal."
"This is such a blatant exploitation of the law, circumvention of the law, that the growing bipartisan consensus is a product of," Curbelo said about the shift after years of inaction by majority of congressional Republicans on gun legislation.
Curbelo's comments follow those of House Speaker Paul Ryan, a Wisconsin Republican who signaled openness Thursday to looking into bump fire stocks, and other Republicans have shown an openness to at least discussing a ban on the devices.
In the wake of the Vegas shooting, Ryan told Hugh Hewitt in an interview for his MSNBC show that "clearly that's something we need to look into."
Ryan's comments come as one senior House GOP member told CNN enough House Republicans are likely to agree to a push for legislation that would ban bump stocks that something could pass in the GOP-controlled House.
"There's enough Republicans who are looking for something that they can say 'I did something.' And the bump stock is an Obama policy. That was approved by the ATF in 2010 and 2012, so why would we defend that?"
The ATF said in 2010 that it did not believe bump stocks to be regulated under existing gun laws because it is a firearm part.
This source believed that there would be a GOP-sponsored bill from a group of moderate Republicans that would narrowly address the issue. While many in the GOP conference will oppose it, this member believed based on conversations with colleagues that there would not be a huge pushback to block it, even from the National Rifle Association.
In fact, the NRA announced Thursday that it supports a review of bump fire stocks
to see if they are in accordance with federal law.
"The National Rifle Association is calling on the Bureau of Alcohol, Tobacco, Firearms and Explosives (BATFE) to immediately review whether these devices comply with federal law," the NRA said in a statement. "The NRA believes that devices designed to allow semi-automatic rifles to function like fully-automatic rifles should be subject to additional regulations."
It's unclear how the legislation would be drafted, but the easiest way would be to simply reverse current ATF policy rather than institute a ban on the manufacture and sales of the accessories.
In his comments, Ryan said that he -- like a lot of members of Congress -- just learned about the accessories in recent days.
"Look, I didn't even know what they were until this week, and I'm an avid sportsman," the Wisconsin Republican said in a clip of the interview that aired Thursday. "So, I think we're quickly coming up to speed with what this is. Fully automatic weapons have been banned for a long time. Apparently this allows you to take a semiautomatic and turn it into a fully automatic so clearly that's something we need to look into."
Asked if GOP leaders will back the bill, Curbelo said "I think leadership needs to see that there is strong support and that's my goal."
He said he's received calls from "dozens" of his House Republican colleagues but declined to say how many would sign on and said he hoped to introduce it sometime on Thursday.
The Florida Republican said his proposal would be a straight ban on bump stocks: "no one can have them, no one can make them, no one can transfer them." His approach mirrors a measure introduced by House Democrats on Wednesday.
Pressed about those gun rights advocates who argue this approach could be a slippery slope to additional gun restrictions, Curbelo waved that argument off, saying, "Do the right thing every time. It's obvious that this is a flagrant circumvention of the law, and no member of Congress should support any circumvention of existing law."
The Senate's No. 2 Republican, Sen. John Cornyn, told reporters Wednesday that he was interested in having a hearing on the matter, a change of tone for a Republican conference that has been dubious about making any regulatory changes related to the Second Amendment.
For his part, President Donald Trump talked about bump stocks in "early stages" with members of Congress traveling with the President on Wednesday, White House Press Secretary Sarah Sanders told reporters during the briefing. She reiterated discussions about bump stocks are something that the White House welcomes.