McAllen, Texas (CNN) Jeb Bush makes a stop near the U.S.-Mexico border in McAllen, Texas, on Monday, just days after Democrats railed the Republican presidential candidate for defending his use of the term "anchor babies" while talking about illegal immigration.

Bush -- seeking a reputation as a moderate Republican when it comes to immigration policy -- has hopes to help woo Hispanic voters. But Democrats are pouncing on his use of the term -- which many find offensive -- to paint him as disingenuous.

Hillary Clinton joined Democratic groups and some Latinos in condemning his comments last week.

"How about 'babies,' 'children,' or 'American citizens,'" Clinton tweeted, responding to Bush getting testy with a reporter and asking the press to give him a "better" word to use.

And Texas Democratic Party Chairman Gilberto Hinojosa said in a statement Monday that Bush's use of the term is "disgusting" and that he "owes million of U.S. citizens, Latinos, and immigrants an apology."

Many Hispanic Republicans, however, aren't concerned about Bush losing momentum. In conversations with more than half a dozen Hispanics who have an influential voice in Republican politics, almost all of them agreed that the term is incredibly offensive, but none considered it a setback for the former Florida governor.

"I think he should have said just children or babies, but the fact of the matter is, I think he pushed back hard on the fact that it's not about rhetoric, it's about policy," said Republican strategist Leslie Sanchez.

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Bush first used the term during a radio interview Thursday morning, calling for "greater enforcement" to address the issue of people coming into the country illegally to have children -- or "anchor babies, as they're described," Bush said -- who become U.S. citizens automatically due to birthright citizenship protections.

Given Bush's close connections to the Latino community -- his wife is from Mexico, he speaks fluent Spanish, he's written a book on immigration and he lives in the Miami area -- it was surprising to hear Bush use the phrase. But he defended his word choice, telling reporters the following day that he didn't regret it.

"What I said is that it's commonly referred to that. I didn't use it as my own language," he said. "You want to get to the policy for a second? I think that people born in this country ought to be American citizens."

The debate over birthright saw a rush of renewed attention after GOP front-runner Donald Trump called to repeal it. Trump has also used the term "anchor baby" in recent days.

Asked to respond to the use of the term by Bush and Trump, Florida Sen. Marco Rubio didn't attack his rivals directly but made sure to avoid uttering the controversial phrase.

"Ultimately, they are people. They are not just statistics. They are human beings with stories," Rubio, also a Republican presidential candidate, said on CNBC last week.

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But Sanchez, who worked in the George W. Bush administration, argued that Hispanic Republican voters ultimately care more about policy than a "blip on rhetoric."

"Democrats want to continue to put the focus on Bush and contrast him, but the truth is Bush is strong on putting down cogent policy solutions that can make a difference," she said.

Rev. Samuel Rodriguez, who invited Bush to speak at the National Hispanic Christian Leadership Conference earlier this year in Houston, condemned the word with passion, calling it "reprehensible."

"No one should be using the phrase 'anchor baby,'" he said. "These are babies created in the image of God."

But Bush has become known as the "standard bearer" of immigration reform among Republicans, Rodriguez said, adding that Bush likely had "zero malice whatsoever" in using the phrase.

"I do believe his heart is pure," he said.

Rachel Campos-Duffy, national spokesperson for the conservative group LIBRE Initiative, which works on outreach to Latino voters, agreed that the narrative of Bush being anti-immigrant won't break through.

"If people are talking about his use of the term and whether or not Bush is a racist, it's obviously ridiculous," she said. "We all heard it. Jeb Bush does not have a racial problem with Hispanics."

Echoing the sentiments of many of those interviewed for this story, Campos-Duffy, who's personally supporting Wisconsin Gov. Scott Walker, said Republicans should avoid allowing "themselves to be pulled into these extreme arguments about birthright" and focus instead on solutions that can bridge Republicans and Democrats to achieve comprehensive reform.

Alfonso Aguilar, executive director of Latino Partnership at the conservative leaning American Principles Project, described last week's headlines over the term as an "educational moment" and doesn't anticipate high profile figures using that word again.

"There are other words, you know, like the N-word that was widely used before and people stopped using that," he said. "People realize how offensive those words are."

The only candidates he sees using it in the future are those who question birthright citizenship or support ending it, and Aguilar, who also worked in the George W. Bush administration, doesn't expect them to get very far with the Latino base.

Candidates like Trump, Texas Sen. Ted Cruz, Louisiana Gov. Bobby Jindal, and retired neurosurgeon Ben Carson have supported the idea of curbing birthright citizenship to children who are brought over illegally, while Walker has offered varying answers on his position in the past week.

"If they were to become the Republican nominees, they're not going to have a shot with Latino voters and that means they're not going to make it to the White House," Aguilar said. "Not because of the term 'anchor baby' but because they want to end birthright citizenship."

But he conceded that "educational moment" will also likely extend to Bush.

"Yeah," he said. "Jeb is probably not going to use it again."