When the U.S. Senate race in Arkansas heated up this summer, Mark Pryor found himself under attack from his opponent with a nasty — and inaccurate — ad claiming that the Democrat had supported giving Social Security benefits to people who had forged identities to work in the U.S. illegally. In Georgia, Democratic candidate Michelle Nunn has been fending off charges that she is “pro-amnesty.”

And here in New Hampshire, Sen. Jeanne Shaheen saw her reelection race tighten after Republican Scott Brown launched a barrage of ads faulting her 2010 vote for the Dream Act, which would have granted legal status to some young immigrants.

Earlier this month, Obama acceded to pleas from a number of vulnerable Democrats to delay until after the November election his promise to use executive power to transform the nation’s immigration system. Though the delay angered some activists, many Democrats in tight races were relieved, hoping that his announcement would cool some of the heat of an issue that could energize the GOP base, particularly in states with low numbers of Latino voters.

But Republicans insist that immigration remains a potent issue in many contested Senate races. The president, they note, merely postponed his threat to use his executive power, and could well grant legal status to as many as several million people now here illegally. Though it is Republicans who have stalled immigration reform in the House, they believe Obama’s delay has given them a new opening to attack Democrats — for addressing issues affecting Latinos only when it is politically convenient. Potentially at stake is control of the Senate, which Republicans will seize if they gain six seats.

Days after the White House announced the delay, Brown laced into Obama and Shaheen in his primary night victory speech in New Hampshire, faulting their “failed policies on immigration” for the surge of unaccompanied minors who came across the border from Central America. (He did not mention that a law encouraging unaccompanied minors to seek refuge in the U.S. passed under President Bush, a fellow Republican.)

“A nation without borders is not a nation at all,” Brown said as he previewed his case against Democratic incumbent Shaheen in a state where a mere 3.2% of the population is Latino. “In Washington, what are they doing? They’re only inviting more chaos at the border by creating amnesty.”

“You have someone before you who will do everything in my power to secure our borders,” Brown said to cheers in Concord, “to make sure that you, and everybody else, is safe and secure when you travel around our country.”

Those kinds of lines are playing well for Republicans in competitive Senate races across the country, where the midterm electorate is typically more white and conservative than in presidential years.

Some Republican strategists fear that the hard line adopted by Republicans such as Brown and Tom Cotton, who is running against incumbent Pryor in Arkansas, could further alienate the GOP from Latino voters, who are key to their hopes of regaining the White House in 2016.

But demographics are on their side this year. Latinos make up 5% or less of eligible voters in eight of nine keenly watched Senate races: Alaska, Arkansas, Georgia, Iowa, Kentucky, Louisiana, Michigan and North Carolina. The lone exception is Colorado, where 14.2% of eligible voters are Latino, making it the one contested state where Obama’s delay in fulfilling his promise could actually hurt the Democrat, incumbent Mark Udall.

“Certainly in a place like Colorado, it could make a difference, partly because there are some Hispanic leaders who have said maybe Latinos should not turn out to vote,” said Mark Hugo Lopez, director of Hispanic Research at the Pew Research Center.

Udall’s campaign said the senator was disappointed by Obama’s decision not to act on immigration before the election. His opponent, Republican Cory Gardner, has charged that Udall and Obama are looking at the issue through “the lens of politics,” but he has been far more careful in his critical statements than some GOP candidates in other states. (In explaining his decision, Obama said that before he would act he wanted to ensure the public “understands what the facts are on immigration.”)