Final approval could come toward the end of the week.
After weeks of negotiation, a bipartisan group of senators on Tuesday night announced the text of their anti-gun violence compromise -- a mix of support for mental health and school security services and changes to some gun laws, including strengthening the ban on domestic abusers possessing firearms and providing incentives to expand the federal background check system for prospective buyers under 21.
The legislation, which the Senate was set to take up for an initial procedural vote on Tuesday night, could quickly pass the upper chamber, according to leaders from both parties.
President Joe Biden, who had previously called for a more robust gun control law -- including an outright ban on assault-style weapons that Republicans rejected -- said earlier Tuesday he would wait to comment on the bill until after it was introduced.
Senators have been eager to preserve momentum on their agreement, which was initially announced in broad strokes on June 12 and sparked by the Uvalde, Texas, school mass shooting last month.
Congress has not taken significant legislative action on guns in some 30 years and conservative lawmakers have long been loath to work on the issue, arguing gun control is ineffective and often unconstitutional. But the Uvalde killings -- following many other such massacres in recent months and years -- spurred another round of talks between Democrats and Republicans in the evenly divided chamber.
The end result, as negotiated by Democrat Chris Murphy and Republican John Cornyn, of Connecticut and Texas, as well as Arizona's Kyrsten Sinema and North Carolina's Thom Tillis, "is not going to please everyone," Cornyn said in a floor speech Tuesday before the bill text was released.
"Some think it goes too far. Others think it doesn't go far enough. And I get it," he said. "But the nature of compromise and the nature of actually wanting to get a result requires that everybody try to find common ground where we can."
Murphy -- perhaps the Senate's most vocal proponent of gun control, representing the site of the 2012 Sandy Hook school Elementary School shooting -- said in his own remarks after Cornyn that he believed they had made important progress.
It was "not window dressing," Murphy said.
"This bill is going to save lives," he continued. "This bill is going to save thousands of lives. This is going to be something that every single member of the Senate who votes for it can be proud of."
The procedural vote on Tuesday night is just the first step in a multi-day process to pass the bill. Exact timing for a final vote is not yet known but could happen toward the end of the week -- with all involved hopeful to approve the legislation before the two-week July 4 recess.
The first vote is to begin debating the legislation. Due to some maneuvering from Democratic leadership, proceeding to debate only requires a simple majority. Still, it could prove a bellwether for initial support for the legislation. At the same time, the just-released bill continues to circulate and many senators haven't yet had time to read it.
It's also unclear when the package's total cost will be released.
Both Senate Majority Leader Chuck Schumer and Senate Minority Leader Mitch McConnell said they supported the bill, indicating that it could well get closer to a super-majority from the Senate, barring more developments. Ten Republicans initially signed onto the June 12 framework, though the draft on Tuesday was announced only by the four key negotiators.
"I support the bill text that Sen. Cornyn and our colleagues have produced," McConnell said in a statement. "For years, the far left falsely claimed that Congress could only address the terrible issue of mass murders by trampling on law-abiding Americans' constitutional rights. This bill proves that false."
In a separate statement Tuesday, Schumer celebrated the group's work and said he would "put this life-saving legislation on the Senate floor for a vote, with an initial procedural vote as soon as tonight and, following that, we will move to final passage as quickly as possible."
The National Rifle Association, however, voiced its opposition to the "overbroad" bill that "falls short at every level."
"This legislation can be abused to restrict lawful gun purchases, infringe upon the rights of law-abiding Americans, and use federal dollars to fund gun control measures being adopted by state and local politicians," the group said. "This bill leaves too much discretion in the hands of government officials."
Key provisions in the legislative text focus on the so-called "boyfriend loophole" in a previous federal ban on convicted domestic abusers having guns and in expanding federal background checks for 18-to-21-year-olds who want to purchase a firearm.
In their speeches on Tuesday -- and perhaps reflecting the triangulated politics of the deal they had reached together -- Cornyn and Murphy sometimes described those new measures and other pieces of their bill in different ways.
As ABC News previously reported, senators had been stuck during the drafting process on, among other things, two points: how to craft language to close the "boyfriend loophole" and how to implement funds to incentivize anti-violence programs.
In his floor remarks Tuesday, Cornyn explained how, in his view, lawmakers worked their way out of those problems.
Regarding the anti-violence programs: According to Cornyn, funds intended to incentivize states to implement so-called "red-flag" laws to remove guns from people deemed to be a danger to themselves or others would be able to be applied to any number of anti-violence programs.
It will be up to states to decide what sort of program they want to use the funds to implement, be it a "red flag" law, mental health courts, veterans courts, outpatient treatment programs and more, Cornyn said. To qualify for federal funds, any of these programs will have to have "robust due process protections," he said.
"We are not going to introduce a national 'red flag' law but we are providing availability of law enforcement-related grants to crisis intervention programs whether you've adopted a 'red flag' program or not, perhaps you've chosen something different. This grant program will give every state funding that implements programs that they themselves have adopted," Cornyn said.
On the loophole, Cornyn said the senators agreed to expand the definition of a boyfriend so that other types of intimate partners convicted of domestic abuse are barred from gun ownership.
He said the bill includes language that allows those convicted of non-spousal misdemeanor domestic abuse to have their gun ownership rights restored after five years if they have a clean record.
"This is an incentive, I think for people who have made a mistake, committed domestic violence and received a misdemeanor conviction, to straighten up their act and not repeat it," he said.
He also gave details on how additional parts of the bill would be implemented.
With the expanded federal background checks for 18-to-21-year-olds, states would control what information they are willing to share with the system. But Cornyn said the legislation "provides an incentive" for states to upload juvenile records.
The legislation also includes funds to "harden" school security and expand the nation's mental health apparatus.
Murphy followed Cornyn on the Senate floor to outline the bill from his perspective and he expressed confidence that the finish line was near, saying, "I believe that this week we will pass legislation that will become the most significant piece of anti-gun violence legislation Congress will have passed in 30 years. This is a breakthrough. And, more importantly, it is a bipartisan breakthrough."
"This was a hard road to get to this compromise, but nothing worthwhile is easy. And nobody in a compromise gets everything they want. This bill will be too little for many. It'll be too much for others. But it isn't a box-checking exercise," he said.
While Cornyn suggested that funding for "red flag" laws could also be applied to some number of anti-violence programs, Murphy emphasized those anti-violence program options were "narrow."
"As Sen. Cornyn said, we will also make those dollars eligible for a narrow range of other court-based anti-violence interventions, something that was very important to our Republican colleagues," Murphy said. Still, this provision could allow states to opt entirely out of "red flag" laws, choosing other anti-violence measures instead.
Murphy also said that allowing those who are domestic abusers to get their right back to purchase a gun will only apply to first-time offenders and have other requirements.
"This bill is going to make sure that no domestic abuser can purchase or own a gun. We are closing the 'boyfriend loophole,'" he said. But he acknowledged: "To be consistent with state felony restoration rights, this legislation will allow individuals to be able to get their rights back after a period of time, but only for first-time offenders and only if there are no crimes of violence in the intervening time."
Murphy did not mention, as Cornyn did, that states will have the ability to choose how much juvenile record information is shared with the national background check system for those ages 18 to 21 seeking to purchase a weapon.
"What we know is the profile of the modern mass shooter is often in the 18-to-21-year-old range," he said. "So this bill has enhanced background checks ... including a call to the local police department, a process that can take up to three days, up to 10 days if there are particular signs of concern that investigators need to perform follow-up on. That enhanced background check is going to make sure that younger buyers who are in crisis have another check performed, perhaps a short period of time in between their decision to buy a lethal weapon to perform a crime and their ability to get that weapon."
Despite the capitulations from Democrats that are already starting to become clear on this bill, Murphy touted it as a major step.
"This week we have a chance to break this 30-year period of silence with a bill that changes our laws in a way that will save thousands of lives. It is a compromise. It is a bipartisan compromise. It is a path forward to the way that both Republicans and Democrats can work together to address some of the most vexing, most difficult challenges this nation faces," he said.
ABC News' Ben Gittleson contributed to this report.