Policies that bar New Orleans law enforcement agencies from inquiring about residents’ immigration status sparked a political tit-for-tat this week, as U.S. Sen. David Vitter criticized Mayor Mitch Landrieu and Orleans Parish Sheriff Marlin Gusman over the rules, and the mayor returned fire by comparing the Republican senator to presidential candidate Donald Trump.

Beyond the political fracas, however, lies a concern by at least one federal official that one local policy might do more harm than good.

The New Orleans Police Department’s policy, which went into effect this week, explicitly restricts officers from asking about immigration status unless there’s a life-threatening situation or those inquiries are necessary to execute warrants or other court orders. Police Superintendent Michael Harrison, in explaining the move, said immigration is a federal, not a local, issue.

The plan was approved by the department’s federal consent decree monitors and was praised by immigration advocates, who say the NOPD’s past willingness to assist with federal deportations has strained relations between police and the Hispanic community.

Gusman’s 2013 policy similarly bars his deputies from investigating inmates’ immigration status. It stemmed from a lawsuit filed on behalf of two men who were held in jail on immigration detainers long after their legal release dates. Gusman also refuses to honor immigration detainer requests at the city’s jail, except for inmates accused of murder and other egregious crimes.

But Vitter, whose repeated calls for tougher border control escalated during his failed bid for governor last year, took aim at Landrieu and Gusman over the policies Wednesday, saying the changes would attract undocumented immigrants to the city who may have criminal histories.

It was his second time criticizing Gusman; Vitter also lambasted him shortly after the jail policy was announced.

“If you do not think creating a sanctuary city will attract more illegal immigrants — some who could be dangerous criminals — you are incredibly naive,” Vitter wrote in a letter to both local officials.

The term “sanctuary city” is often applied to U.S. cities with policies designed to protect undocumented immigrants from deportation, despite a federal program that uses local law enforcement to help identify those immigrants. There are more than 300 such cities around the country, and they’ve been widely decried by congressional Republicans, who pushed unsuccessfully last year to block their federal funding.

Vitter’s letter, however, did little to dissuade Landrieu.

“The NOPD’s new bias-free policing policy is consistent with federal law and was designed in partnership with the U.S. Department of Justice, federal consent decree monitors, the U.S. Department of Homeland Security and local Immigration and Customs Enforcement staff,” Landrieu said.

“Be clear: NOPD will continue to work with federal immigration officials when there is a warrant or following a violent crime, but their priority is going after violent criminals, not raiding churches and markets to intimidate immigrant communities,” he wrote.

Landrieu also drew parallels between Vitter and Trump, whom the mayor also has publicly denounced.

“Like Donald Trump, David Vitter wants you to believe that all immigrants are criminals,” Landrieu said, adding that Vitter employed similar “divisive, race-baiting politics” when he ran for governor.

Landrieu said the NOPD’s policy was enacted via a federal partnership.

However, the sheriff’s policy concerns U.S. Deputy Secretary of Homeland Security Alejandro Mayorkas, who expressed reservations in a recent interview.

“I would posit to the city of New Orleans: If we’re dealing with somebody with, for example, multiple felony convictions and they’re here illegally, why release them to the street and why force us to deploy our fugitive operations team to apprehend someone? Why not turn them over?” Mayorkas said. “The community doesn’t want this individual roaming the streets.”

Mayorkas also expressed optimism that all city agencies would one day begin cooperating with federal immigration officials again, pointing to Los Angeles as “a prime example of a major county that was not cooperating” but has since signed on to the government’s Priority Enforcement Program.

“We’ll have to work with New Orleans to see what type of relationship we can rebuild here,” Mayorkas added.

Jim Mustian contributed reporting.

Follow Jessica Williams on Twitter, @jwilliamsNOLA.