Latinos mainly blame Republicans, who control the House and have buried the Senate bill, but they also have soured on Mr. Obama. The federal government has so aggressively enforced existing immigration laws that one national Hispanic leader recently nicknamed the president “deporter in chief” for allowing nearly two million people to be deported.

A day after that widely reported gibe in Washington, at Denver’s Spanish-language radio station KBNO (“Que Bueno” to its audience), the host Fernando Sergio devoted his three-hour talk show to asking listeners whether they agreed with the criticism, or “has President Obama done the best he can against Republican opposition?”

“The majority were very angry at the president,” Mr. Sergio said in an interview at the station, where pictures of John and Robert Kennedy, Bill Clinton and Mr. Obama hang on the walls. “People feel like he’s made some promises that he hasn’t fulfilled, that he can do more” — like expand his 2012 order that deferred deportations of young people brought to the country as children, a group known as Dreamers.

“If I were a Democratic consultant,” Mr. Sergio added, “I would have been concerned.”