None of the legislators mentioned the NRA's call for armed guards at school as an option under consideration.
Obama has not ruled out issuing executive orders on some gun control measures to enforce laws already on the books, such as bolstering the way gun sales are tracked.
The president reiterated his desire on Monday for more robust background checks for gun buyers, keeping high capacity magazines away from criminals, and a ban on assault weapons.
"Will all of them get through this Congress? I don't know," Obama said. "But what's uppermost in my mind is making sure that I'm honest with the American people and members of Congress about what I think will work, what I think is something that will make a difference."
Working with Congress will be paramount in curbing gun violence, Thompson said, singling out a ban on high capacity magazines as an example of a measure that could garner Republican support. A full-scale assault weapon ban would be tougher to pass the GOP-controlled House, he argued.
Obama also said on Monday that the gun lobby was "ginning up" fears the federal government will use the Connecticut tragedy to seize Americans' guns. At least part of the frenzy is little more than marketing, he implied.
"It's certainly good for business," the president said, responding to a question about a spike in weapons sales and applications for background checks since the December killings.
Biden has said he's found widespread support for universal background checks and restrictions on the sale of high capacity magazines, which gun control advocates believe contribute to more bloodshed at mass shootings.
The influential NRA, among other gun rights groups, has vowed to fight any new gun restrictions -- like an assault weapon ban.
Gun control advocates, gun violence victims, the NRA, video game makers and others have met with the Biden-led task force.
In New York, Democratic Gov. Andrew Cuomo on Tuesday signed into law a series of new gun regulations -- the nation's first since the Newtown shootings.
Both New York's GOP-controlled Senate and Democrat-controlled Assembly approved the measure by overwhelming margins.
It includes a statewide gun registry and adds a uniform licensing standard across the state, altering the current system, in which each county or municipality sets a standard.
Residents are also restricted to purchasing ammunition magazines that carry seven bullets, rather than 10.
"The changes in New York are largely cosmetic," said CNN legal analyst Paul Callan, who described existing regulations as "the toughest gun laws in the United States."
Lawmakers in at least 10 other states are reviewing some form of new gun regulations in the new year.
Meanwhile, new national polls indicated a majority of Americans support some or most gun control measures.
By a 51%-45% margin, Americans questioned in a new Pew Research Center poll said it was more important to control gun ownership than to protect gun rights.
And by a 52%-35% margin, a new ABC News/Washington Post survey indicates the public says it is more likely to support some forms of gun control after last month's massacre. However, the polls showed continuing divisions on political and gender lines.