The National Rifle Association “does not want terrorists or dangerous people to have access to weapons,” said Jennifer Baker, a spokeswoman for the organization’s lobbying arm. “But this is a constitutional issue,” she said, adding that mere suspicion should not be enough to take away the right to own a gun.

While federal gun control legislation has gone nowhere in recent years, certain states have had more success. Connecticut has passed some of the strictest gun laws in the country, including measures enacted after the 2012 massacre at Sandy Hook Elementary School in Newtown, when a gunman killed 20 schoolchildren and six staff members before killing himself.

Daniel Webster, the director of the Johns Hopkins Center for Gun Policy and Research, said it was unclear what the practical implications of Connecticut’s proposed ban would be in stopping someone who is determined to carry out an act of terrorism. That person could simply travel to another state.

“Seems to me that the greatest importance of this is to get the ball rolling so more people follow, and ideally the federal government,” Mr. Webster said. “I suspect more states will do this.”

Devon Puglia, a spokesman for Mr. Malloy, said the governor’s office was “working with the federal government” to gain access to the watch lists and determine which would be most appropriate to use. The governor, he said, would sign the order as soon as he received approval to use the watch lists for that purpose.

The federal government’s terrorism watch list is a database maintained by the Terrorist Screening Center, an arm of the Federal Bureau of Investigation. It fuses information gathered from many sources — including law enforcement, military and intelligence agencies, as well as foreign governments — and is used for various purposes, including keeping certain people off planes or blocking certain noncitizens from entering the country. The no-fly list is a subset of the watch list.

The debate has become fodder for presidential candidates, with most Republicans attacking the idea of denying gun sales to people on the no-fly list.