Gun deal negotiators may finish draft of new law this week, ahead of speedy vote, Sen. John Cornyn says

Now that a tentative deal on trying to curb gun violence has been reached in the Senate, negotiators are working to make the proposal a reality -- and fast.

Sen. John Cornyn of Texas, the lead Republican on the gun talks, said Monday he would like to see lawmakers finish the bill's legislative text sometime this week ahead of what Democrats have said would be a quick vote on the finalized bill.

"My hope is that we can complete that job in the next few days, hopefully by the end of the week, so that the bill will be available for all senators -- indeed all the world -- to read," Cornyn said on Monday in a lengthy floor speech.

Such a timeframe would set up a possible vote on the Senate floor next week.

Senate Majority Leader Chuck Schumer, D-N.Y., vowed Monday to hold a vote "quickly" once the legislative text was finalized, but acknowledged there was still more that needed to be done before a bill reached the floor. The framework of the agreement has 10 Republicans in support -- enough to avoid the threat of a GOP filibuster that has stymied past gun laws. But Republican aides told ABC News the bipartisan deal was on the principles and not the details, which are still being worked out.

"Make no mistake about it, we have a lot of work left to do before we actually pass a bill. But yesterday's announcement was a positive and necessary step in the right direction," Schumer said.

A group of 20 senators -- 10 Democrats and 10 Republicans -- announced on Sunday they had reached a broad agreement after working for weeks following the mass shooting at an elementary school in Uvalde, Texas, that left 19 young children and two teachers dead.

The outline of the deal includes funding for mental health and school safety; incentives for states and localities to pass "red flag" laws to take away guns from those deemed a danger to themselves or others; and strengthening the federal background check system, especially for potential gun owners under the age of 21 and for people convicted of domestic violence -- closing the so-called "boyfriend loophole."

Cornyn emphasized Monday that the proposed reforms would not add any more restrictions upon "law-abiding gun owners" but said he believed they will "save lives."

"This is not an easy debate," Cornyn said. "It's emotional. It can be divisive. But it is also very important that we act."

The agreement doesn't include everything Democrats, including President Joe Biden, have called for in the wake of the latest onslaught of mass shootings across the U.S. This weekend alone there were at least 10 such killings.

In a nationwide address earlier this month, Biden urged Congress to ban assault-style weapons -- which were previously outlawed in the 1990s and early 2000s -- and high-capacity magazines, repeal immunity for gun manufacturers and more.

"We spent hours with hundreds of family members who were broken, whose lives will never be the same," Biden said then. "They had one message for all of us: Do something."

ABC News White House correspondent Karen Travers asked White House press secretary Karine Jean-Pierre on Monday if the tentative gun deal delivered on what those parents said they wanted.

"What he heard and all of you may have heard from the folks in the community, as well, is to do something," Jean-Pierre replied. "The president has called on Congress to do something. They are doing something."

Biden wants to see the gun safety reform bill "on his desk to sign as soon as possible," Jean-Pierre added.

Cornyn himself admitted the delicacy of delivering on anti-gun violence legislation in such a closely divided Senate.

"Most often we hear people say, 'Do something.' Well, they don't give you a lot of guidance on what that something looks like -- and when you begin to dig down into the details, you find out there is not a lot of consensus about what that something should look like," he said in his speech on Monday.

He said he believed focusing on keeping guns away from "criminals and people with mental health problems" was a winning formula.

"I'm hoping that 10 Republicans supporting the bill is not a ceiling but is the floor," he said.

If passed, the deal would be the first major piece of gun control legislation to make it through Congress in three decades.

"I urge my colleagues to think of all the lives we can now save by turning this framework into law," Schumer said on Monday. "Americans have waited long enough for us to take action. Too many lives, too many have been already lost."