WASHINGTON — Americans appear open to a softer immigration policy than Republican presidential candidates have preached in the past. Even a majority of Republicans in the country can see themselves voting for someone who would protect millions of immigrants from deportation, according to a new poll.

The Associated Press-GfK poll finds that Republicans overwhelmingly want a candidate to reverse President Barack Obama’s unilateral action to postpone deportations. But most could see themselves voting for someone who would keep that policy in place.

The poll was conducted before Democrat Hillary Rodham Clinton’s effort this week to magnify the difference between letting people who are in the country illegally have legal status, which some Republicans support, and letting them gain citizenship, which she supports. The poll suggests that people don’t see much difference.

“This country has always had open doors,” said Dean Talmadge, a Republican from suburban Seattle. “I don’t just have a problem with immigration, as long as they are here, working and following the rules.”

That’s a sentiment shared by most Americans, who according to the poll, favor a path to citizenship by 53 to 44 percent. It did not make a significant difference if respondents were asked instead whether they supported a path to legal status short of citizenship: 50 percent said they favored it and 48 percent opposed.

Clinton’s remarks Tuesday were aimed at Hispanic voters, who heavily supported Obama in 2012, especially in battleground states such as Florida, Colorado and Nevada, where Latinos form an influential bloc.

She also said she would expand Obama’s executive action, which shielded many young immigrants brought to the country as children, along with parents of citizens and permanent residents, from deportation.

In the poll, about 6 in 10 Americans and 4 in 10 Republicans support those key measures of the Obama policy. Americans overall were split about evenly on whether they would prefer to vote for a candidate who would keep or undo the executive action.

“I don’t have a problem if they want to earn their citizenship while they are here,” said Darlene Harmison, a Republican from Colfax, Iowa, a rural town east of Des Moines. “I’m not saying send them all back. But I do agree they need to speak English.”

Most GOP presidential prospects support allowing people who are in the country illegally to stay while pursuing legal status. But they stop short of offering citizenship, although a few have not ruled that out as an eventual prospect.

Their positions, which are sometimes in flux and not always fleshed out, seem to reflect what the poll finds to be fragile acceptance among Republican voters of policies that had spelled trouble for past candidates.

Three-quarters of Republicans in the poll say they would prefer to vote for a 2016 candidate who would undo Obama’s immigration action — even more than the two-thirds of Republicans who say they oppose a path to citizenship or legal status.

Even so, 55 percent of Republicans either prefer a candidate who would support the essence of Obama’s policy or say that they could vote for someone who does if they agreed on other issues. Even among conservative Republicans, nearly half — 47 percent — could at least imagine voting for a candidate who would keep it in place.

In 2006, Republican Sen. John McCain, who supported a bipartisan bill that included a path to citizenship, was surprised at the backlash he received from Republicans in early voting states as he began taking steps toward running for president.

“There’s no question it was not well received by certain segments of our electorate,” said David Roederer, McCain’s 2008 Iowa campaign director.

More recently, Florida Sen. Marco Rubio, a GOP presidential candidate, faced pushback from tea party supporters when he co-authored legislation that would have set out a path to citizenship for those who met certain conditions, over time. The bill passed the Senate, failed in the House and convinced Rubio that immigration policy would have to be changed only piecemeal.

He no longer promotes a citizenship path but, like former Florida Gov. Jeb Bush and some other rivals, supports legal status as a possibility.

Even those viewed as hardliners are not calling for mass deportation of the millions in the country illegally, signaling a possible shift in the GOP political landscape as the nation — judging by the poll and other measures — also evolves.

The AP-GfK Poll of 1,077 adults was conducted online April 23-27, using a sample drawn from GfK’s probability-based KnowledgePanel, which is designed to be representative of the U.S. population. The margin of sampling error for all respondents is plus or minus 3.4 percentage points. Some questions were asked of a half sample and have higher margins of error.

Respondents were first selected randomly using phone or mail survey methods and later interviewed online. People selected for KnowledgePanel who didn’t otherwise have access to the Internet were provided access at no cost to them.