The list of sanctuary cities has grown to more than 340, and they shielded an average of 1,000 immigrants a month from deportation last year — and more than 2,000 of those released have been arrested for yet more crimes, according to a report released Thursday by the Center for Immigration Studies.

Among those released are illegal immigrants accused of murder and brutal assaults, said Jessica Vaughan, the report’s author.

The report was released just as the Senate was poised to begin debating legislation to crack down on sanctuary jurisdictions.

Indeed, hours after the report was released, Senate Majority Leader Mitch McConnell, Kentucky Republican, set up an early test vote this month on a bill that would crack down on sanctuaries. The proposed legislation would take away some federal grant money and distribute it to counties and cities that cooperate with federal authorities.

Sanctuary cities exploded onto front pages in July, when a Kathryn Steinle, a 32-year-old woman, was killed while walking with her father in San Francisco, and an illegal immigrant released by the county was charged with the shooting.

Ms. Vaughan said San Francisco ranked eighth worst on the list of offenders after releasing 252 immigrants last year whom federal officials asked to be held.

Worst of all was the Santa Clara County Jail in California, which released 1,349 immigrants that U.S. Immigration and Customs Enforcement agents had asked to be held for custody transfer. Los Angeles and Santa Rita jails were tied for second worst, with 572 releases each.

“Our elected officials must not sit back and watch these sanctuary jurisdictions continue to release thousands of criminal aliens back into our communities in defiance of ICE efforts to deport them, and then witness the harm that inevitably ensues when these removable offenders strike again,” Ms. Vaughan said.

All told, 9,295 immigrants whom ICE wanted to pick up from January through September last year were released instead.

Of those, 5,947 had significant criminal records, Ms. Vaughan said, citing an internal ICE report.

ICE agents were able to track down about 30 percent of them. Of the others, 1,377 have been arrested in connection with crimes committed since their release.

ICE also confirmed to Ms. Vaughan that 340 jurisdictions are refusing to cooperate to some extent with federal deportation requests.

The Center for Immigration Studies keeps a map and tally of those jurisdictions and recently added nine names to its list. They include Prince George’s County and Montgomery County in Maryland and Chesterfield County in Virginia. The map is available at CIS.org/Sanctuary-Cities-Map.

Immigrant rights advocates defend sanctuary cities, saying deportation is a job for the federal government, not local jurisdictions. The advocates also say that when local police do cooperate with ICE, it strains relationships with Hispanic communities in particular, whose members then fear reporting serious crimes.

The conflict has left Homeland Security Secretary Jeh Johnson in the middle. Bowing to legal challenges, he scrapped the Secure Communities program that mandated state and local prisons and jails to hold illegal immigrants.

Instead, he has taken a voluntary approach through the Priority Enforcement Program, which asks communities to cooperate and promises to ask for only the most serious criminal offenders.

On Wednesday, Mr. Johnson said 13 of the 25 biggest sanctuary localities have expressed interest in the Priority Enforcement Program.

“More are coming online, and I expect we will reach agreement with major cities in the near future,” the secretary said in a briefing on the state of immigration enforcement, delivered to the Congressional Hispanic Caucus Institute.

Mr. Johnson said sanctuary cities’ refusal to cooperate has hurt his deportation efforts, and it’s one reason why deportations in fiscal year 2015, which ended Oct. 1, are at their lowest level in a decade, down nearly 50 percent from their peak in 2012.

From January through June, the secretary said, localities released more than 16,000 illegal immigrants whom his agents wanted to be held for deportation.

Ms. Vaughan said the Priority Enforcement Program is a poor substitute because it allows localities to control the process.

Some jurisdictions, such as Los Angeles, have agreed to participate but under strict conditions they set, meaning hundreds of illegal immigrants who could be deported will be shielded instead, she said.

The effect of sanctuary cities is huge. If officers can pick up a target from a jail, they can schedule several in a day, and the process takes only a few people. If authorities have to pick up a target who has been released, it requires locating the person, a full team to conduct hours of surveillance, a case officer to handle paperwork — and the potential for a violent public confrontation.

The House has passed a bill to crack down on sanctuary cities, and Senate Republicans have introduced their own version this week. Mr. McConnell has scheduled an Oct. 20 vote to test whether Democrats will filibuster.

The legislation would require Mr. Johnson to publicly list sanctuary cities, would withhold some federal grant money from those jurisdictions, and would distribute that funding to cities, counties and states that cooperate with federal immigration authorities.

It would also create a mandatory five-year minimum prison sentence for any illegal immigrant found guilty of sneaking into the U.S. twice or sneaking back even once after a felony conviction.

Ms. Vaughan said that type of legislation is the best step Congress could take to have an immediate effect on illegal immigration.

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