Republican and Democratic Senate supporters of a new gun bill, from left, Lindsey Graham, Bill Nelson, Susan Collins and Kelly Ayotte. The measure would ban sales to terrorism suspects.

After several gun control measures were defeated in the Senate this week, a bipartisan group of senators have called for banning gun sales to terrorism suspects on the government’s “no-fly” list, according to this story from the New York Times.

The push is being led by Sen. Susan Collins (R ME) and Sen. Heidi Heitkamp (D-ND) along with Sen. Tim Kaine (D-VA), who has been mentioned as a potential running mate for Hillary Clinton in the 2016 Presidential race.

According to the story, the Collins proposal is “tailored narrowly to prohibit gun sales to suspected terrorists who appear on the government’s ‘no-fly’ list or its ‘selectee’ list, which requires more rigorous security checks before a person is allowed to board an airplane.”

The story says those combined lists include about 109,000 people, of whom 2,700 are American citizens. This is in contrast to the federal terrorist screening database, which includes about a million names and was the focus of Sen. Diane Feinstein’s (D-CA) measure that was defeated on Monday.

The Collins measure assuages a chief complaint about using the no-fly list to bar citizens from exercising a constitutional right, often voiced by gun rights advocates, in that it allows for an appeal by any citizen or green card holder blocked from making a firearms purchase because they are on one of the lists. The measure would also award lawyer’s fees if the appeal is successful. Currently, there is no way to appeal inclusion on the no-fly or selectee lists.

However, the Collins measure doesn’t address the other chief criticism of the entire concept of using terrorist lists for gun control: there is still no due process for a person’s inclusion on such lists. It is, instead, an arbitrary and secret process. The lists themselves are also not available to the public.

Sen. John Cornyn (R-TX) said he is apprehensive about the provision in the Collins proposal because it permits an appeal only after a gun sale is prohibited, the story says.

“I think it’s a slippery slope when an American citizen is denied a constitutional right without forcing the government to come forward with some evidence on the front end, as opposed to leaving that on the back end,” Cornyn said in the story. “But we’ll see how the vote comes out.”

Collins’ bill would also require notification of federal and local law enforcement agencies if anyone who had been on the lists in the previous five years seeks to buy a weapon. The story says this provision is meant to directly address the recent attack in Orlando.

To win approval, the measure would need the support of 60 senators. The story says it isn’t currently clear that enough Republicans would back it. Three of the 54 Senate Republicans appear to now, and if that doesn’t change, the vote would be 16 short of the number needed for bill to pass, even if the entire Democratic conference votes in favor of the measure—which isn’t guaranteed, the story says.