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Jim Schick, 52, gave a blunt assessment while he waited for Mr. Romney to appear at a campaign stop in the Cleveland suburbs. “I think the cat is out of the bag,” Mr. Schick said. “You can’t send young people back to Mexico.”

Mr. Schick said the immigration issue could be neutralized for Republicans if Mr. Romney taps Sen. Marco Rubio of Florida, a Cuban-American conservative, as a running mate and embraces an immigration policy that aims to ensure illegal immigrants are paying taxes and working to become legal citizens, while deporting criminal illegal immigrants.

“It is inhumane to think that you are going to send those people back,” Mr. Schick said, alluding to young adults. “They are in our schools. They are pledging allegiance to the flag, just like everybody else.”

Dialing down the rhetoric

Mr. Romney did not, as some GOP operatives privately hoped, try to pound home a claim that the president’s order is illegal and violates the Constitution.

Romney advisers think that would needlessly restrict his actions once in the Oval Office and is unnecessary at this point in the campaign. For the first time in many decades, more voters say that deciding what to do with illegals already here is more important than border security, and a recent Gallup poll found that only 2 percent of likely voters list immigration as an important issue this election year, down 17 percentage points from a similar Gallup survey in 2006.

In addition, a Bloomberg News poll released Tuesday showed that voters liked Mr. Obama’s move by a margin of more than 2-1. The telephone survey of 734 likely voters, taken from Friday to Monday and with an error margin of 3.6 percentage points, showed 64 percent of respondents favoring the president’s order, 30 percent opposing and 6 percent saying they were not sure.

But even the most outspoken interest groups promoting border security and enforcement of the laws against illegal immigrants say they do not rule out some compromise short of mass deportation for illegal immigrants and their children.

“The Dream Act is a tough one,” said Steven Camarata, research director for the Center of Immigration Studies. “If Romney does something like the Dream bill, it’s not necessarily a winner.”

Mr. Obama’s move may “help marginally with liberal voters and Hispanics,” Mr. Camarata said. “But how much help is questionable — 69 percent of the Hispanic vote, up from 67 percent, [while] at same time alienating white voters in the Midwest.”

“That’s not to say we could never support a limited form of Dream Act,” Mr. Camarata said.

Ralph Z. Hallow reported from Washington.