The White House gave a cool welcome on Monday to Democratic legislation that would effectively ban online or mail-order purchases of ammunition in the aftermath of the mass shooting at an Aurora, Colorado, movie theater.

"I haven't seen the specific piece of legislation that has been offered up today," spokesman Josh Earnest told reporters at the daily press briefing. "But as that and other pieces of legislation make their way through the legislative process, we'll evaluate them."

The proposal, crafted by Democratic Senator Frank Lautenberg and Democratic Representative Carolyn McCarthy, aims to restrict the ability of Americans to buy unlimited quantities of ammunition over the Internet, or by mail order, anonymously.

President Barack Obama has called for a common sense response to the slaughter in Aurora. But the White House has played down his appetite for new legislation as opposed to tightening or toughening existing measures—such as background checks—to keep firearms out of the hands of criminals or the mentally ill. And the president has underlined his support for the Second Amendment to the Constitution.

"He believes in the second amendment of the constitution, in the right to bear arms," Earnest said again Monday. "But he also believes that we should take robust steps, within existing law, to ensure that guns don't fall in the hands of criminals or others who shouldn't have them.

The new legislation, dubbed the Stop Online Ammunition Sales Act, rests on four pillars, according to Lautenberg's office:

It requires anyone selling ammunition to be a licensed dealer.

It requires ammunition buyers who are not licensed dealers to present photo identification at the time of purchase, effectively banning the online or mail order purchase of ammo by regular civilians.

It requires licensed ammunition dealers to maintain records of the sale of ammunition.

It requires licensed ammunition dealers to report the sale of more than 1,000 rounds of ammunition to an unlicensed person within any five consecutive business days.