Washington (CNN) -- A Senate committee approved legislation on Tuesday that would expand background checks covering all U.S. firearms sales, part of a federal gun-control push prompted by December's school massacre in Connecticut.
The Judiciary Committee decision by a party-line vote of 10-8 cleared the way for the Democratic-crafted background check measure to be debated and put to a vote in the full Senate.
Less clear is the fate of a proposed ban on military assault rifles proposed by Sen. Dianne Feinstein, a California Democrat. The panel put off a vote on that plan, but is expected to revisit it as early as Thursday.
There is little expectation Feinstein's proposal, which President Barack Obama backs along with expanded background checks and other steps, will win congressional approval due mainly to stiff opposition from lawmakers aligned with the politically potent National Rifle Association.
The background check proposal by Sen. Charles Schumer, a New York Democrat, is also expected to face tough opposition once the full Senate takes it up.
It is among the first gun-control proposals to begin moving through Congress since the school massacre in Newtown, Connecticut, by a lone gunman armed with an assault weapon jolted the nation and renewed debate on the issue.
A bipartisan effort in the Senate in recent weeks to put together background check legislation crumbled prior to the committee vote. One sticking point concerned checks involving private sales and what records would be maintained once guns are purchased.
Schumer's bill would mandate background checks for all gun sales, including private transactions.
It would also require increased cooperation by states with the FBI's National Instant Criminal Background Check System (NICS) and would prohibit people deemed by the courts as unfit to own guns from obtaining them.
In opposing Schumer's measure, Sen. Chuck Grassley disputed whether tougher background check laws would prevent criminals from obtaining a gun.
"Obviously, criminals do get guns. They obtain them because they do not comply with background checks," the Iowa Republican said. "Supporters of this bill contend that if we only make background checks universal, criminals will not get guns. But criminals do not comply with existing background check laws."
Schumer took issue with Grassley's comments.
"On terrorism, on robbery, on murder, on money laundering, we never see the argument that we shouldn't have laws because the bad people will get around them anyway," a visibly frustrated Schumer said. "Only on this issue and it makes no sense."
Whether the measure approved by the committee on Tuesday can garner enough support in the Senate remains to be seen. Finding Republican support has been tough.
For weeks, Schumer was in talks with Oklahoma Republican Sen. Tom Coburn in an attempt to find agreement across the aisle. Hours before the committee took up the issue last week, those talks fell apart.
Lawmakers once supportive of Schumer's bipartisan effort, including Sen. Mark Kirk, an Illinois Republican, and Sen. Joe Manchin, a West Virginia Democrat, quickly distanced themselves from it when it became clear the legislation wasn't what they envisioned.
The NRA has said increased checks are nothing more than an attempt to create a national gun registry, a move the group vehemently opposes.
"Criminals won't participate in a 'universal' system. They'll always steal or get their guns and everything else they want on the black market. Reasonable people know that criminals will never be part of the 'universe,'" NRA Executive Vice President Wayne LaPierre said on the group's website.
Schumer said in a statement following the committee meeting that the issue of background checks remains the political "sweet spot" on gun control and promised continued efforts to find bipartisan support.
A second measure was approved on Tuesday by the panel that would establish a framework for the Department of Education to set safety guidelines aimed at protecting schools from gun violence.
The bill, proposed by Sen, Barbara Boxer, a California Democrat, was approved 14-4 with bipartisan support.
Before the committee vote Sen. Lindsey Graham, a South Carolina Republican, said he hoped for increased discussion on funding armed security officers in schools but voted in favor of the measure promising to negotiate details when it reaches the Senate floor.
Last week, the Senate panel also approved a bill that would toughen laws on gun trafficking and straw purchases of firearms.
CNN's Kevin Bohn contributed to this report