Most Americans say they back a plan that would allow certain illegal immigrants to stay in the country, but support for the idea slips when President Barack Obama's name is attached to the question, according to a recent Reuters/Ipsos poll.

The results underscore the polarized nature of the debate over how to overhaul immigration laws and how to address the estimated 11 million people living illegally in the country.

Congress has been unable to agree on reforms for years. In 2014, Obama bypassed the Republican-controlled Congress to issue an executive order to shield more than 4 million undocumented immigrants from deportation and offer them work permits, but he was blocked by court challenges from Republican-governed states.

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The Supreme Court is set to rule by the end of June on whether Obama had the power to make the change. The decision will come just weeks before Republicans choose their nominee for November's presidential election.

Republican front-runners Donald Trump and Ted Cruz have called for a crackdown on illegal immigration, with Trump promising to build a wall along southern U.S. border with Mexico.

But Hispanic voters are an increasingly powerful voting bloc, and the debate over Obama's immigration action could hurt Republican attempts to make inroads into the important demographic.

The Reuters/Ipsos poll was taken in the week following the Supreme Court's decision to hear the immigration case.

The poll shows 61 percent of Americans support the plan to relax immigration policy for some undocumented people when it is described in general terms without using Obama's name, including 42 percent of Republicans. Half of Republicans opposed the idea.

But when the same plan was described as being an executive action taken by Obama, support fell to 54 percent overall, with only 31 percent of Republicans supporting it and 62 percent opposing the measures.

For Democrats, 78 percent supported the plan when it was described without using Obama's name, and 80 percent supported it when the president's name was attached to it.

The online poll of 1,200 people has a credibility interval of 3.2 percentage points, including 5 percentage points for Democrats and 5.4 percentage points for Republicans.