Separately Monday, the head of a major gun control group said they are not pushing to ban the AR-15-type weapon used in the Orlando massacre because their top priority now is passing the watch list legislation and an update to the Brady background check law that was enacted in 1993.
The Senate push comes in the wake of the shooting tragedy at a nightclub in Orlando that killed 49 and wounded scores of people. It's not clear, though, that the restriction would have blocked that shooter, Omar Mateen, from purchasing the weapons he used in the attack since he may not have been on that watch list.
The same legislation, authored by Sen. Dianne Feinstein, D-California, failed on a largely party line vote in the Senate seven months ago. Republicans argued that many people end up on the watch list mistakenly and they didn't want to see those people lose their constitutional right to bear arms because of a bureaucratic snafu.
In an emotional speech on the Senate floor, Senate Minority Leader Harry Reid vowed to act quickly to pass the Feinstein legislation.
Senate Democrats "are going to, as soon as we can, force vote on this terror loophole. We're going to do this as soon as possible. There is no excuse for allowing suspected terrorists to buy guns," Reid said.
Congressional Democrats are also pushing for votes on legislation that would ban gun sales to individuals that have been convicted of misdemeanor hate crimes. Rhode Island Democratic Rep. David Cicilline, who has openly gay, unveiled a bill recently to add this new restriction, and Pennsylvania Democratic Sen. Bob Casey introduced the companion measure on Monday.
One senior Democratic leadership aide told CNN that top Democrats feel that these two targeted measures -- one focused on the terror watch list and one on hate crimes -- have a better chance of moving this year.
"We know a Republican Congress isn't going to do an assault weapons ban," the aide said.
On a conference call with reporters, Sen. Chuck Schumer of New York and other Democrats said while they are interested in passing other gun control measures -- such as universal background checks and a ban on assault weapons -- they want to focus now to prevent suspected terrorists from retaining their rights to buy weapons.
"In the wake of Orlando, we have to think about kind of country and what kind of Senate we're going to be. Are we going to take the painfully obviously commonsense steps and make sure that terrorists can't get guns or are we going to bow down to the (National Rifle Association) so that suspected terrorists can continue to get their hands on guns?" Schumer said.
Schumer said they may push for a vote as an amendment to a funding bill for Justice Department programs that is on the floor now although it's not clear Republicans would agree, especially so soon after the December roll call vote when the measure was defeated.
Republicans also point to a counter proposal from Sen. John Cornyn that got more votes than the Feinstein measure did even though it failed to get the 60 votes it needed to pass. That proposal would have put a three-day delay on a gun purchase by someone on the watch list but also give the attorney general the ability to take the issue to court to get a permanent ban.
Speaking in Cleveland, Clinton said she supports blocking gun purchases by those being investigated by the FBI and also banning assault weapons.
"If the FBI is watching you for a suspected terrorist links, you shouldn't be able to just go by a gun," the presumptive Democratic presidential nominee said. "And you shouldn't be able to exploit loopholes and evade criminal background checks by buying online or at a gun show."
Florida Democratic Rep. Alan Grayson, whose district is near the location where the attack took place, called on both the House and Senate to "ban assault rifles permanently."
Grayson is campaigning to replace GOP Sen. Marco Rubio in the next Congress.
House Democrats are planning to call for immediate action on gun control measures on Tuesday, but in the GOP-led chamber, there are no plans to move bills adding new restrictions on gun purchases.
In a separate press conference call, Dan Gross, the president of the Brady Campaign to Prevent Gun Violence, argued that his organization is more interested in keeping guns out of the hands of people who shouldn't have them than banning certain types of guns outright.
"It hasn't been our priority policy push for a few years because we are focused on what we can to prevent the most possible gun violence and we are certain the best way to do that is not by focusing on taking away certain guns away from all people but what we agree on which keeping all guns away from certain people -- convicted violent criminals, domestic abusers, the dangerously mentally ill," Gross said.