[Read:Butterflies, Bulldozers and Barbed Wire Rally Border Wall Opponents]

New Mexico’s governor, Michelle Lujan Grisham, announced last week that she had ordered a partial withdrawal of National Guard troops from her state. “New Mexico will not take part in the president’s charade of border fear-mongering by misusing our diligent National Guard troops,” Ms. Grisham, a Democrat, said in a statement.

Early Monday, Gov. Gavin Newsom of California announced that his state, too, would withdraw National Guard troops sent earlier to help the federal government boost security at the border.

In Nogales, Ariz., the City Council on Wednesday passed a resolution condemning the recent installation of new barbed wire along the existing border wall in that city, calling it an “indiscriminate use of lethal force” that is “typically only found in a war, battlefield or prison setting.”

Gaining public support for the idea of a wall at an event such as the rally in El Paso will be important for Mr. Trump, as talks for a bipartisan agreement on border security appeared to have stalled on Sunday amid lingering disagreement over how much should be spent on a border barrier. President Trump, who initially proposed spending $25 billion on a wall, now is looking for $5.7 billion. Democratic lawmakers have talked about a figure closer to $1.3 billion to $2 billion.

A second government shutdown could be in the works if no agreement is reached.

Bob Giles, a car dealer from Lafayette, La., who is building a new Volvo dealership in El Paso, said that he understood how Mr. Trump’s proposed wall could seem like a good idea from an “outside perspective.”

“But when you spend time in El Paso and talk to people, it’s clear they take offense that a wall made a difference in the city,” said Mr. Giles. While Mr. Trump often speaks of a crisis on the border, Mr. Giles said he hasn’t seen one.

“That’s not the word I would choose,” he said. His own hope is to sell cars not just to Americans, but also to business executives from Juárez who have homes in El Paso. “This is an incredibly vibrant market with lots of construction going on,” he said. “It makes a difference when you see it with your own eyes.”