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A child holding a sign asking cars at a stoplight to honk for immigration reform drew a decent stream of beeps, but also a number of head shakes and plenty of cars whose drivers ignored the rally altogether.

Mr. Goodlatte won re-election last year with 65 percent of the vote in the district and is not expected to be in danger next year, either.

That underscores part of immigrant rights advocates’ problems in swaying House GOP members — so many House Republicans are in safe districts where their bigger problem is a challenge on the right rather than losing to a Democrat running from the left.

In interviews and at a town hall he held this month, Mr. Goodlatte ruled out creating a “special pathway” to citizenship for illegal immigrants — but Mr. Gutierrez said that could leave some middle ground if it means reopening some of the paths that used to exist in federal law, such as allowing people to readjust their status from within the U.S.

“I would have hoped he would have found time to be here so he could explain to us just that,” Mr. Gutierrez told reporters. “I want to understand it. I don’t want to reject it. I don’t want to reject any avenue that leads to us resolving the problem.”

Asked whether he was harming his cause by these visits, Mr. Gutierrez said he has insulated himself by taking on both sides, including criticizing his party leader, Mr. Obama.

The congressman has even been arrested protesting outside the White House, accusing the president of deporting too many people and not putting enough effort into winning an immigration agreement. He regularly points out that Mr. Obama is poised to deport his 2 millionth immigrant before the end of this year.

“When I go to the White House, I already know what seat to take when I go to the Roosevelt Room. I take the seat farthest away from where the president sits,” the congressman said of the regular meetings the Congressional Hispanic Caucus has had with Mr. Obama. “That’s OK. That’s the way things work in Washington, D.C.”