Mayor Freddie O’Connell of Nashville would rather be talking about the state of the sidewalks. New traffic signals. Even the increase in the property tax rate. Instead, he has been busy addressing the fallout from a round of federal immigration raids last month in his liberal-leaning city.

First, angry residents accused city officials of helping federal agents detain more than 100 people during the raids, which Mr. O’Connell, a first-term Democrat, quickly denied. After the outcry, he ordered city departments to let his office know about any outreach from immigration agents; he also expressed support for a community fund that seeks private donations for immigrant families.

Tennessee Republicans then demanded investigations into whether the mayor had violated state law. The state has banned local governments from adopting “sanctuary city” policies, which it describes as limiting cooperation with immigration enforcement and giving undocumented immigrants “the right to lawful presence.”

Mr. O’Connell drew more Republican wrath when his office published the names of some immigration agents who had contacted the city. He has maintained that the names were published accidentally because they were in public records, including summaries of emergency calls.

The city has since removed the names and denies violating state law. Nonetheless, two congressional committees are investigating the effect of Nashville’s policies on federal immigration enforcement.

The backlash demonstrates how Mr. O’Connell, 48, is caught between the laws of his deeply Republican state and the progressive leanings of many of his constituents. It is perhaps the biggest test yet of his cautious pragmatism, at a moment when local leaders are on the front lines of the Trump administration’s aggressive overhaul of federal policy.