By BRIAN FALER | 2/14/14 1:53 PM EST

'This is a taxpayer fraud issue,' Lindsey Graham says. | AP Photo

Republicans looking for budget savings are targeting a politically delicate group: illegal immigrants.

Some want to crack down on quirks in the law that allow undocumented workers to claim the child tax credit, among the government’s biggest tax breaks, and one that can result in the IRS cutting checks to beneficiaries.

The push, which could raise more than $19 billion over a decade, comes even as Republicans face pressure this election year to expand their appeal to a growing bloc of Hispanic voters. Indeed, the effort is being led in the Senate by some of the party’s biggest advocates of immigration reform. Democrats are crying foul and accuse Republicans of being hostile to Latinos.

“Name one Republican who’s been more forward-leaning on immigration than myself,” said Sen. Lindsey Graham (R-S.C.), one of the architects of the bipartisan immigration overhaul that cleared the Senate in June. “This is a taxpayer fraud issue. You’re not advancing the cause of immigration reform by allowing $19 billion to be stolen from the Treasury.”

Republicans have repeatedly pushed the plan this year, first to pay for an extension of emergency jobless benefits. When that was rejected, Sen. Kelly Ayotte (R-N.H.) offered it this week to defray the cost of repealing a $7 billion cut in cost-of-living adjustments (COLA) to military pensions. Senate Majority Leader Harry Reid (D-Nev.) blocked her proposal from a vote, and lawmakers agreed to instead finance the pension cut repeal by extending across-the-board budget cuts known as sequestration.

“They use this tax benefit to provide for their children’s basic needs,” said Sen. Barbara Boxer (D-Calif.). “Why should any of us attack children — literally take food out of the mouths of children?” She called the proposal “anti-immigrant.”

But Ayotte promises to continue to press the issue.

“It needs to be addressed regardless of whether we do it on this military COLA bill,” she said. “I will offer it again.”

There will likely be plenty of opportunities with lawmakers facing demands to pay for extending a package of expired tax cuts, avoid looming reductions in Medicare payments to doctors and deal with a highway trust fund slated to run dry as soon as this year.

Republicans have been pushing similar proposals in the House.

The effort comes in the face of warnings that Republicans must expand their outreach to Hispanic voters, a group Republican presidential candidate Mitt Romney lost by more than 40 percentage points in 2012.

Asked how their effort might appear to Hispanics, Graham said: “This is not an injustice to say to illegal immigrants that we’re not going to give tax credits to people who are in illegal status — I don’t think that’s an injustice.”

Ayotte, who also backed the immigration overhaul, said she wants to treat undocumented workers “fairly or I wouldn’t have supported immigration reform.” She said, “that doesn’t mean I don’t want to address fraud.”

At issue is a tax break created in 1997 to help families with the cost of raising children. The credit, which has been repeatedly expanded by both parties, now offers up to $1,000 per child under the age of 17. It’s refundable, which means people can get money back from the government even if they don’t owe any taxes. It’s projected to cost about $57 billion this year.

Because of idiosyncrasies in the law, illegal immigrants are able to claim the credit.

About $4.2 billion was paid to unauthorized workers in 2010, up from $924 million in 2005, according to a report by the Treasury Department’s Inspector General for Tax Administration.

Undocumented workers are not allowed to work in the U.S., though, if they do, the law requires them to pay taxes.

“The law is crazy,” said Sen. Dick Durbin (D-Ill.).

That’s because the IRS requires workers to file tax returns regardless of their immigration status. They must have Social Security numbers, though those ineligible to receive them can instead get a nine-digit individual taxpayer identification number to use for tax reporting purposes. About 3 million tax forms filed in 2010 had these ITINs.

Republicans want to require those claiming the refundable portion of the child credit to have Social Security numbers. Ayotte initially proposed requiring the filers themselves to have numbers. She later revised her plan to only require the children who are claimed to have Social Security numbers, not the filers, in response to Democrats’ complaints that her bill would have penalized the children of illegal immigrants who themselves are American because they were born here.

“I have addressed many of their concerns,” said Ayotte.

That cut the projected savings to $19.6 billion over a decade, down from about $27 billion, according to the Joint Committee on Taxation, though still more than enough to finance an extension of jobless benefits anticipated to cost $6 billion.

The savings comes from excluding illegal immigrants from the program.

Republicans note that Congress has already agreed, as part of the 1996 welfare overhaul, to require those claiming the government’s biggest refundable credit, the earned income tax credit, to have Social Security numbers. And they point out that Democratic President Bill Clinton had called for that change in his budget that year.

Democrats say there are better ways of finding savings in the government’s budget.

“Taking money away from children of low-wage, tax-paying families is indefensible,” said Durbin. “Those who are undocumented come forward and pay their income taxes, and naturally expect to be treated like anybody else who pays income taxes.”

Illegal immigrants are more than making up for the child tax credit by the money they pay into Social Security, through that program’s payroll tax.

By law, they’re not allowed to claim Social Security benefits. But in 2010, unauthorized workers contributed $12 billion to the retirement program, according to Social Security’s chief actuary, Stephen Goss.

Asked about those taxes, Ayotte said: “I’d have to look at the numbers.”