Washington (CNN) -- Caught off guard by the Obama administration's shift in immigration policy last week, Republicans on Tuesday refined their response in an effort to lessen any political bounce for the president at what had been a tough time in his campaign.
Certain GOP presidential nominee Mitt Romney led the Republican counterattack, accusing President Barack Obama of undermining the possibility of long-term immigration reform by taking the partial step of halting deportations of some young illegal immigrants who came to America as children.
Speaking in an interview on Fox News Radio, Romney said Obama could have proposed immigration reform earlier in his term, when Democrats controlled both chambers of Congress, but instead waited until his re-election was pending.
"The reason this came out was the president is trying to shore up his base with Latino voters," Romney said, noting that Obama's campaign has struggled in recent weeks to deal with a disappointing jobs report and what he called self-inflicted wounds.
"He is also trying to change the subject from his miserable speech last week, from his gaffe that the private economy is doing fine," Romney added.
House Speaker John Boehner, R-Ohio, acknowledged that Obama's move "puts everyone in a difficult position," telling reporters it will "make it much more difficult for us to work in a bipartisan way to get to a permanent solution."
Boehner also said that the policy change might violate the Constitution and that the president made no attempt to work with Congress.
Senate Majority Leader Harry Reid, D-Nevada, called such Republican arguments hypocritical and "phony outrage."
"The complaints are varied, but they have one thing in common: None of them actually take issue with the substance of President Obama's directive," Reid said, adding: "They just don't like the way President Obama made the decision -- or that he will get the credit for bringing out of the shadows 800,000 trustworthy young men and women who know no other home but the United States."
Tuesday's spin by leading Republican moderates Romney and Boehner differed from the initial reaction by conservatives who called Obama's move an amnesty -- a buzzword for right-wing opposition that has stymied immigration reform efforts.
Now the Republican criticism focuses on the process rather than the content, and for good reason.
A Bloomberg poll released Tuesday showed that a solid majority of likely voters back the policy change Obama announced Friday, including coveted independent voters.
According to the survey, 64% of respondents agree with the new policy, while 30% disagree. Among independents, the percentage in agreement rises to 66%, while 26% are opposed.
The poll also showed that 86% of Democrats support the new policy, while 56% of Republicans disagree with it.
Wendy Schiller, a political science professor at Brown University, called Obama's move "very shrewd" because it showed leadership on an issue important to his liberal base and the Hispanic community.
"This is a major, major step toward cementing the bond between the Latino community and the Democratic Party in an active way," Schiller told CNN, adding that voters will remember that the Democrats were "the party that let me stay in this country, ... that kept my family intact."
In announcing the change Friday, Obama said it would make U.S. immigration policy "more fair, more efficient and more just."
The new policy allows people younger than 30 who came to the United States before the age of 16, pose no criminal or security threat and were successful students or served in the military to receive a two-year deferral from deportation. They also can apply for work permits.
Administration officials emphasized that the change offers no path to citizenship or legal residency, and they said it will allow the Department of Homeland Security to focus resources on keeping out new illegal immigrants and tracking down repeat offenders, criminals and others.
The policy change mimics legislation called the DREAM Act, which Democrats were unable to push through Congress in 2010 because of a Republican-led filibuster in the Senate.
In that vote, five Democrats in the Senate joined Republicans in blocking the measure, while three GOP senators voted with Democrats to bring it up. The final tally of 55-41 was five votes shy of breaking the filibuster.
Reid complained Tuesday that Republicans have continually resisted efforts to work with Democrats on immigration reform and the DREAM Act
"Every time Democrats offer to work together on comprehensive immigration reform, Republicans find an excuse to fight sensible change," Reid said. "And every time Democrats propose bipartisan legislation to provide a pathway to citizenship for children brought here illegally through no fault of their own, Republicans find an excuse to oppose our practical reforms."
After Obama announced the policy change Friday, conservative House Republicans immediately blasted it as a backdoor amnesty, with Iowa Rep. Steve King threatening to sue the administration.
However, Republican aides acknowledge that immigration is a divisive issue within their party, with GOP members in competitive districts needing to appeal to Hispanic voters to win re-election this fall.
To Schiller, the Obama move put Republicans in a tough spot if they oppose his policy of halting the deportations.
Noting the children of illegal immigrants affected by the change had to be very young when they arrived in America with their parents, she called it "a population that is impossible to paint in a negative light."
"How can you support a policy that would break up families? You can't," she said.
Boehner and Romney reflected that reality Tuesday.
"I think we all have concerns for those who are caught in this trap through no fault of their own and are here," Boehner said.
Romney pledged to "work from the beginning of my administration to put in place a piece of legislation which deals with this issue on a long-term basis," saying young illegal immigrants who serve in the military "should be able to become permanent residents of the United States," a step short of full citizenship.
In the Senate, Minority Leader Mitch McConnell, R-Kentucky, avoided a direct answer to repeated questions from reporters about his response to Obama's move, deferring instead to a speech by Romney later this week at a major Latino conference in Florida.
"He is the leader of our party from now until November and we hope beyond, and we're gonna wait and hear what he has to say about it," McConnell said.
Romney has endorsed efforts by Florida Sen. Marco Rubio, who has been mentioned as a possible running mate, to draft a Republican version of the DREAM Act.
Obama's move effectively ended that effort, a Rubio aide acknowledged Monday.
"We're re-evaluating our plans," Rubio spokesman Alex Conant said. "The president's announcement took away our momentum and made the politics a lot tougher."
Rubio had planned to introduce the bill this summer, citing concerns about certain students facing deportation ahead of the new school year, but his office said the president's step erased any sense of urgency.
"This approach requires a long-term solution, and my biggest concern -- and I may be proven wrong -- my biggest concern is that by doing it this way, the president had undermined and set back the hopes of getting a long-term solution done on this issue," Rubio told CNBC on Tuesday.
Audrey Singer, a senior fellow at the Brookings Institution who specializes in immigration issues, said the exceptionally bitter political divide right now made Republican calls for comprehensive immigration reform ring hollow.
During the GOP primary campaign, Romney said he wanted illegal immigrants to deport themselves, "meaning if we make it miserable enough for (them), they will leave," Singer told CNN.
Now his desire for comprehensive reform also sounds like campaign talk that might not translate to results should Romney win the White House in November, she said.
If Obama's policy change is close to what Rubio was proposing, then both sides should be able to coalesce around it, Singer noted, adding: "The fact is that Obama has enacted something similar that doesn't somehow add up to bipartisan support, and that, my friend, is politics."
Both Obama and Romney are scheduled to address the National Association of Latino Elected and Appointed Officials annual conference later this week in Florida, a key general election battleground state with a large Hispanic population.
Both candidates also released Spanish-language ads this week. Romney unveiled a spot Monday attacking the president's record on the economy, while the Obama campaign released a spot Tuesday featuring Latino talk show host Cristina Saralegui, who endorsed the president after last week's policy shift, according to the campaign.
CNN's Gabriella Schwarz, Jessica Yellin, Jim Acosta and Ashley Killough contributed to this report.