A group of nine senators on Tuesday unveiled a bipartisan compromise aimed at keeping guns out of terrorists' hands in the wake of the June 12 Orlando shooting.

The legislation spearheaded by Sen. Susan Collins, R-Maine, would prohibit people on the government's no-fly list and the government's selectee list used for extensive screening at airports from purchasing a gun.

The bill would allow U.S. citizens and green-card holders to appeal a decision blocking his or her purchase of a firearm, and if successful, to be awarded attorney's fees. It also contains a look-back provision so that if anyone has appeared in the government's broad terrorism database over the last five years, the FBI would be immediately notified if they attempt to buy a gun.

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"Our goal is simple and straightforward: We want to make America safer," Collins said at a press conference unveiling the proposal, which she said "would help keep guns out of the hands of terrorists."

The total number of people on the watch lists under the legislation totals 109,000, Collins said, and most of them are foreign nationals.

Sen. Tim Kaine, D-Virginia, is among the nine senators who are backing this legislation.

"I'm sick of the shootings. I'm sick of the vigils. I'm sick of the homicide victim support groups. I'm sick of the claims that we'll do something about it. I'm sick of the partisan rhetoric, and I'm really sick of getting to the end of all of it and not doing something about it and seeing that happen again and again and again," Kaine said at the press conference as he explained why he supports the measure.

The introduction of the deal comes a day after the Senate rejected four measures that were intended to address the issues of terrorists' ability to buy guns and background checks. Each of those proposals was considered to be partisan, though there were a few members from each party who defected on the measures.

Kaine feels that there will be no meaningful gun control reform unless it is bipartisan.

"We've got to make progress and in order to make progress, we've got to do it in a bipartisan fashion," he said.

Sen. Lindsey Graham, R-South Carolina, also supports the proposal. He said he owns an AR-15 himself and he addressed the concerns of some Republicans that the watch lists include people who shouldn't be on them.

"The likelihood of someone being on this list and buying a gun to use it in a terrorist act to me is far greater than the likelihood of an innocent person being on the list," Graham said. "We can fix the problem with the innocent person."

The other senators co-sponsoring the legislation are Sens. Heidi Heitkamp, D-North Dakota; Kelly Ayotte, New Hampshire; Bill Nelson, D-Florida; Jeff Flake, R-Arizona; Angus King, I-Maine; and Martin Heinrich, D-New Mexico.

While it's up to Senate Majority Leader Mitch McConnell, R-Kentucky, to schedule a vote on the proposal, Senate Majority Whip John Cornyn, R-Texas, said that Collins would get a vote on her langauge if she wants one.

These developments on Capitol Hill come just over a week after the Orlando shooting that left 49 people dead and 53 wounded. The FBI had investigated the gunman, Omar Mateen, in 2013 and 2014 and interviewed him several times. He was still able to purchase and use in the shooting an AR-15-style rifle and a semi-automatic pistol.

CBS News' John Nolen contributed.