But not everyone in the Republican ranks is buying that. Particularly those who have no fear of alienating that constituency anytime soon, like Iowa Congressman Steve King, traditionally one of the fiercest critics of illegal immigration in Washington.
Last week, King announced his new bill that would end the practice of birthright citizenship (sometimes referred to as "anchor babies"). The legislation would amend Section 301 of the Immigration and Nationality Act to clarify those classes of individuals born in the United States who are nationals and citizens of the United States at birth.
This is not the first time such a bill would be introduced in the GOP-held House.
Although it has been interpreted that the 14th Amendment would prevent such a law, King previously said he doesn't believe that to be the case.
CBS News reported that King's bill has three co-sponsors: Rep. Phil Gingrey (R-Ga.), Gary Miller (R-Calif.) and Rob Woodall (R-Ga.).
The Los Angeles Times reported that last week, state lawmakers from at least five different states visited Washington D.C. to discuss the possibility of passing similar ordinances in their respective states.
Lawmakers from at least five states said they planned to take the suggested laws back to their legislatures for consideration. Daryl Metcalfe, a Republican state representative from Pennsylvania who founded the group that brought together lawmakers from 14 states, said the laws would help slow down an "illegal alien invasion."
When Danny Verdin, majority whip for the South Carolina state Senate, took the lectern at the news conference to explain his intention to introduce the proposed laws in his state, he raised the coming April commemoration of Confederate troops firing on Ft. Sumter in Charleston, S.C., at the start of the Civil War.
"South Carolina may have been out front leading 150 years ago at Ft. Sumter, but we are happy to work collaboratively on this to cure a malady," said Verdin, a Republican.
As Verdin went on to say that he had dozens of colleagues in South Carolina eager to draw up the bills, a protester shouted, "I cannot ignore this inhumane and racist bill!
King's bill comes as many Republicans are admitting that their harsh rhetoric on undocumented immigrants contributed to Mitt Romney's poor showing with the particularly crucial set of voters. Romney lost the Latino vote by a 69-29 percent margin.
Meanwhile it remains to be seen how serious Republicans (and Democrats) are about working with President Obama on a comprehensive immigration reform package, the usual rhetoric from conservatives that nothing can happen until the border is "secured" may need to be altered a bit.
On Tuesday, the New York Times reported a new study that says immigration control has become "the federal government's highest criminal law enforcement priority."
The 182-page report was an opening salvo in a contentious debate over immigration that President Obama has pledged to lead this year. Its purpose was to marshal publicly available official figures to show that the country has built "a formidable enforcement machinery" since 1986, the last time Congress considered an overhaul of the immigration laws that included measures granting legal status to large numbers of illegal immigrants. Spending on immigration enforcement was 15 times greater last year than in 1986, the report found.
The report responds to lawmakers, mainly Republicans, who have argued that federal authorities must do much more to strengthen enforcement before Congress can consider any legalization for an estimated 11 million illegal immigrants in the country.
"The 'enforcement first' policy that has been advocated by many in Congress and the public as a precondition for considering broader immigration reform has de facto become the nation’s singular immigration policy," the report concluded.