Kate Steinle was shot to death on July 1 on one of San Francisco's busiest piers.
Investigators still are trying to determine how Juan Francisco Lopez-Sanchez, the man accused in Steinle's killing, allegedly obtained the .40-caliber pistol, according to the source.
The Bureau of Land Management said the service weapon was stolen from the officer's vehicle on June 27.
Lopez-Sanchez told CNN affiliate KGO-TV
that he fired the weapon, but it was an accident. He said he'd found the gun wrapped inside a T-shirt before it accidentally went off.
It was not clear whether Lopez-Sanchez understood all the questions posed in English and Spanish during the KGO interview. He appeared disoriented and gave contradictory answers about what had happened and how he felt
Lopez-Sanchez is an undocumented immigrant and a repeat felon who has been deported to Mexico five times
, according to immigration officials.
The case has drawn the attention of presidential candidates and brought a renewed focus on U.S. immigration laws and the role local authorities should play in enforcing them.
The key question is whether San Francisco's policies set the stage for the shooting by putting a criminal on the streets instead of delivering him into the hands of federal authorities who could have deported him again.
In an interview Tuesday with CNN
, Democratic presidential hopeful Hillary Clinton said San Francisco should have listened to the Department of Homeland Security and made a mistake when it didn't send Lopez-Sanchez packing.
"I have absolutely no support for a city that ignores the strong evidence that should be acted on. ... If it were a first-time traffic citation, if it were something minor, a misdemeanor, that's entirely different," she said. "This man had already been deported five times. And he should have been deported at the request of the federal government."
Immigration and Customs Enforcement said Lopez-Sanchez has seven felony convictions, four for drug offenses. His most recent deportation was in 2009.
On Tuesday, Lopez-Sanchez pleaded not guilty to murder and weapons charges. His bail was set at $5 million.