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But in a letter to House Speaker John A. Boehner, Ohio Republican, she stopped short of embracing the Senate’s bill, which signals that the legislation is dead despite Senate Democrats’ calls for the House to pass it as is.

Instead, Mrs. Pelosi called on the Republicans to make sure they touch all areas of the immigration debate no matter how they put bills on the floor.

“If you decide to take up various elements of comprehensive immigration reform under separate votes, it is essential to remember that those key elements are interconnected and necessary for reform,” Mrs. Pelosi said in her letter.

She and other Democrats questioned whether House Republicans could pass any legislation without including a path to citizenship, which is crucial for most Democrats.

In Dallas, Mr. Bush, who repeatedly clashed with his party over immigration during his tenure, again warned Republicans about the tenor of the debate.

“I don’t intend to get involved in the politics or the specifics of policy, but I do hope there is a positive resolution to the debate. And I hope during the debate that we keep a benevolent spirit in mind and we understand the contributions immigrants make to our country,” Mr. Bush said at a ceremony granting citizenship to 20 people at the Bush Institute.

After he failed in efforts to get an immigration reform law in 2006 and 2007, Mr. Bush stepped up enforcement and began a process that added thousands of Border Patrol agents in the Southwest.

Mr. Obama has continued those border efforts, though he has changed deportation priorities to leave most illegal immigrants with little chance of being removed once they settle in the U.S.

Distrust of Mr. Obama and his commitment to the rule of law dominated the Republican discussion.

“This president has shown an astonishing hubris to ignore the law as it’s written,” said Rep. Trent Franks, Arizona Republican.

Some Republicans saw danger in even moving the issue piece by piece. They fear their leaders will send those bills into a House-Senate conference with the massive Senate bill, and what will emerge is a bill that has a pathway to citizenship for illegal immigrants.

Rep. John Fleming, Louisiana Republican, said leaders gave assurances that a final conference report “would accurately reflect the will of the House, the Republican conference.”

But Mr. Fleming said that was too vague and not enough for him. He wanted to hear who the negotiators would be.

“I would feel much better if I actually knew who was going to be on that,” he said.