WASHINGTON — The Orlando, Fla., massacre reignited on Monday the debate over whether Americans like Omar Mateen who have appeared on the government’s terrorist watch lists, or who were otherwise suspected by the authorities of ties to terrorist groups, should be allowed to buy a gun. In Congress, even some supporters of gun rights seemed to be having second thoughts.

Six months after Republicans in Congress defeated a measure that would have closed the so-called terror gap after the San Bernardino, Calif., attack, Senate Democrats moved swiftly on Monday to renew the debate over tightening federal gun laws.

As a first step, the Democrats demanded that Republicans take up legislation aimed at banning the sale of guns or explosives to people who have appeared on watch lists, or who have been suspected by the Justice Department of ties to terrorist organizations.