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“We don’t need another massive, Obamacare-like bill that is full of surprises and dysfunction after it becomes law,” said Rep. Bob Goodlatte, Virginia Republican and chairman of the House Judiciary Committee, who has written some of the piecemeal bills.

Some key Republicans also have said the way Mr. Obama treated House Republicans during this month’s spending and debt debates — in which he refused to negotiate with them over any of their demands — has soured the working relationship.

Some Democrats agree, saying that while Mr. Obama can cheerlead for action from the sidelines, he should not get involved in the negotiations.

On Wednesday, Rep. Luis V. Gutierrez, an Illinois Democrat who has taken the lead on this issue for his party, went to the House floor to invite Republicans to work around the president.

“Those on the other side of the aisle say they do not trust the president and can’t work with him. Well, OK, fine. Then work with us,” Mr. Gutierrez said. “There are 435 of us, we need 218 to pass a bill and the president doesn’t get a vote.”

Complicating matters is the short schedule for the rest of this year. The House likely has only five more weeks of business planned for 2013, and Republicans in the chamber are preoccupied with writing a 2014 budget and fighting the president’s health care law.

That has left immigrant rights advocates arguing that Mr. Obama can take unilateral action.

They want him to expand his 2012 nondeportation order, which grants tentative legal status to young illegal immigrants, to include most of the estimated 11 million illegal immigrants now in the country.

“Our position is that President Obama can actually do something today, right now,” Ms. Franco told The Washington Times.

She said if Mr. Obama took that step, it would force Republicans to confront the issue and could advance the debate.

Mr. Obama has said that while his nondeportation policy for young illegal immigrants was legal, he doubts he has the authority to do a blanket policy for all illegal immigrants. But the activists dispute that.

“It’s very clear that there is a legal way to do this. It’s a question of will he do it politically,” Ms. Franco said. “That’s why we’ve issued this call for nonviolent civil disobedience, focused on ICE, and being willing to name the president as partially responsible.”