WASHINGTON — In many ways, President Obama’s unilateral shift in immigration policy was a bluntly political move, a play for a key voting bloc in the states that will decide whether he gets another term. But as political moves go, it held the potential for considerable payoff.

It sent a clear signal to fast-growing Hispanic populations in Florida, Colorado, Nevada, Virginia and other states that he understood their frustration at his lack of progress so far in addressing problems with the immigration system and reducing the number of deportations.

After two weeks in which his re-election campaign often seemed to be struggling to cope with events and losing the upper hand to Mitt Romney, Mr. Obama, for a day at least, was able to drive the agenda. And the president’s announcement put Mr. Romney, whose party is already split on the issue, in a tough spot, pressuring him to choose between further alienating Latino voters who chafed at the anti-illegal immigration stances he took in the primary season and alienating conservatives who reject policies resembling amnesty.

The timing of the announcement appeared to have been carefully calibrated. Next week, Mr. Romney and Mr. Obama are scheduled to appear before a group of Hispanic elected officials on successive days in Florida, a ready-made opportunity for the president to draw a contrast in their positions before a swing-state audience. Mr. Obama is also scheduled to meet on Monday with a group of immigration activists at the White House. The Supreme Court is expected to rule soon, perhaps Monday, on Arizona’s legislation that requires state law enforcement officers to check immigration status during traffic stops and makes it a crime for illegal immigrants to work in the state.

Especially with unemployment remaining high, the politics of immigration remain sensitive for both parties, with neither Democrats nor Republicans wanting to be seen as jeopardizing jobs for American workers. But polling suggests considerable public opposition in both parties to deporting people already in the country illegally, and support for steps that would allow otherwise law-abiding young people who arrived in the United States illegally to stay. The Obama administration had continued to deport illegal immigrants, including some young people, breaking up families and draining enthusiasm for the president’s re-election among some Hispanic voters.

Democrats have long seen immigration as the best example of how Mr. Romney’s move to the right during the bitterly contested Republican primary could complicate his ability to create a broad general election coalition. During the primary season, Mr. Romney opposed the Dream Act, proposed legislation that would have allowed many young illegal immigrants to remain in the country and would have given them a path to citizenship. As he has moved into the general election and confronted the need to compete for Latino voters, his campaign has tried to finesse the issue by saying that the focus of his outreach to Hispanics would be on jobs and the economy, and his initial response to Mr. Obama’s decision on Friday was to focus less on its substance than its unilateral nature.

Speaking after an event in Milford, N.H., on Friday, Mr. Romney criticized the president’s executive order for making it “more difficult to reach a long-term solution” for dealing with the fate of young people who are in the United States illegally “through no fault of their own.” Mr. Romney did not respond to questions about whether he would repeal Mr. Obama’s order if elected.

In both substance and tone, his remarks were a softening of the posture he struck during the Republican primary fight, when he said he would veto any bill allowing children brought to the country by illegal immigrants to become permanent residents.

They should “return home, apply and get in line with everyone else,” Mr. Romney said in a debate in January.

His position apparently shifted under the influence of Senator Marco Rubio of Florida who has been preparing a Republican version of the Dream Act similar to what the president ordered Friday.

Congressional Republicans were more pointed in their criticism, but they too were careful not to oppose some kind of solution to the problem of young people who are in the country illegally but who are productive, otherwise law-abiding residents. Senator Lindsey Graham, Republican of South Carolina, denounced it as “possibly illegal” for essentially bypassing lawmakers. Mr. Rubio said the announcement would be “welcome news for many of these kids desperate for an answer,” but that by going around Congress, the president had made it “harder to find a balanced and responsible long-term” solution.

Their caution reflected concern within the Republican Party that they are at risk of giving up a chance to win the political allegiance of Hispanics, not just for this election but for years to come. Though the Dream Act was originally co-sponsored by a conservative Republican, Senator Orrin G. Hatch of Utah, in recent years most conservatives have worked to block any changes to immigration policies that would provide a path to citizenship for illegal immigrants, and many conservatives have backed measures to limit the rights of illegal immigrants already in the United States.

But Jeb Bush, the former Florida governor, called this week for the party to do more to connect with Hispanic voters. And just before Mr. Obama’s announcement, former Gov. Haley Barbour of Mississippi told reporters in Washington on Friday that Republicans, for political and business reasons, should support plans to let workers who are not citizens stay in the country.

“I believe America is in a global battle for capital,” Mr. Barbour said. “If you are a worker who has been here for any length of time, we have to have a path, not to citizenship, but a secure knowledge that they will be able to work.”

Many Hispanic leaders had been critical of the president for not doing more to address immigration. But on Friday, they said they welcomed his move, whatever his motivation. “We know this is political — we like that it’s political,” said Robert Meza, a Democratic state senator from Phoenix. “People are smart enough to know that of course it’s politics, but if their agenda moves forward, they’re happy.”

Mr. Romney has warned that if Republicans did not find a better way to appeal to Hispanic voters, “it spells doom for us.”

But his campaign has concluded so far that its outreach should be an economic message, not a modulation of the anti-illegal immigration stances Mr. Romney took during the primaries. His advisers observe that polls show Hispanic voters rank jobs and the economy their most pressing issues, followed by education.

Some advisers have argued that Mr. Romney has more to lose by exposing himself to charges of shifting positions over illegal immigration than he has to gain by appealing to a slice of the Latino constituency that ranks immigration a top concern — because those voters already support Mr. Obama.