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Polls show that Mr. Brown is well-positioned to win the Sept. 9 GOP Senate primary, but that Mrs. Shaheen, a former governor, holds a 10 percentage-point lead over him in a general election matchup.

Still, Mr. Obama’s approval rating has turned negative here, which could cause headaches for Mrs. Shaheen.

“This was always going to be a very competitive race,” said Jim Merrill, a New Hampshire-based GOP strategist. “But I look at this as he is within striking distance. He is right where I would want him to be.”

The idea that New Hampshire has become a front in the battle over immigration policy could strike some as odd given that it sits nearly 2,000 miles from the U.S.-Mexico southern border, where thousands of families and unaccompanied children from Central America are trying to enter into the United States illegally to escape the violence and poverty in their countries.

But political observers say that Mr. Brown is looking to bolster his conservative credentials among Republicans still wary of him, and who call him a “Republican in Name Only,” or RINO.

“I think it is a strategy to get a certain segment that may think that he is more moderate and that segment may get pretty fired up on that issue,” said Neil Levesque, executive director of the Institute of Politics at Saint Anselm College. “It may be a small segment, but if they are fired up to go to the polls on that one issue and he can get their loyalty — he can have success with that.”

James Carville, a Democratic strategist, agreed.

“His strategy is probably that he wants to consolidate his base and branch out more when he gets to the fall,” Mr. Carville said. “He has had a rough go here so far. This has not worked out very well for him, but it is probably something he needs to do to right his ship among core Republicans supporters.”

For his part, Mr. Brown said he “probably” would have opposed the Senate immigration bill and called the Obama administration’s non-deportation policy for “dreamers” — the Deferred Action for Childhood Arrivals program — a “neon light” for the thousands of illegal immigrants trying to travel across the border.

“We need to compassionately make sure [the children] are safe and secure, and we need to put them back into their own countries,” he said.