Mr. Pearce says he wants to fix what he calls a broken immigration system, and he sees himself as a border-state pragmatist.

“One side says everybody just gets citizenship, the other side says deport them or put them in jail, whatever. And I’m saying there is another alternative that would allow families to stay together, that would allow people to work, that just would not make them citizens,” Mr. Pearce said last week in response to a question at a town hall-style meeting in Deming. “Because that makes me very nervous as a policy.”

Later in the evening, he added: “I think that the plan I’m suggesting is the compromise plan. It’s somewhere out between kick them out and give them full citizenship.”

Mr. Pearce’s proposal would create a guest worker program, which would allow illegal immigrants to stay in the country as long as they were working. But it would not provide citizenship to those workers over time. He would allow illegal immigrants to become citizens, but they would first have to return to their home country and wait in line there.

“You can’t get control of the borders if you tell people you can come here illegally and you can work until you work your way to the front of the line,” Mr. Pearce said in an interview on Thursday. “The whole world would want to do it that way. Who would want to wait and do it properly?”

Despite Mr. Pearce’s strong opposition to a path to citizenship, advocates of an overhaul still believe that he might be persuadable, and various groups have descended on his district over the August break, holding marches and protests and filling his public meetings. “They simply realize that this is a 34 percent Republican district, and they’re hoping to bring enough pressure that would change me,” he said.