Lawmakers on Capitol Hill appeared to be reaching consensus this week that immigration reform can be done in pieces — a key procedural concession by President Obama and his fellow Democrats that could help revive chances for a bill next year.

But even as they said they could accept the GOP’s plans to tackle immigration bill-by-bill, rather than one massive measure, Democrats insisted that before the process is done, Congress will have to tackle all parts, including rewriting the legal immigration system to granting citizenship rights to most current illegal immigrants.

“We have always said on that score that the speaker is the speaker, and any way he wants to bring the bill to the floor, in pieces or in big chunks or whatever it is, we just want to see legislation come to the floor so that Congress can act upon that legislation, the House can, and send it to the conference table with the Senate,” House Minority Leader Nancy Pelosi, California Democrat, said.

House Speaker John A. Boehner, Ohio Republican, last week seemed to close the door on chances for an immigration bill to pass this year when he said the House would work on its own timeline and would not enter into negotiations with the Senate, which has passed a single broad bill legalizing illegal immigrants and rewriting the legal immigration system. Mr. Boehner also insisted on a step-by-step approach.

Asked this week if that meant the issue is dead, Mr. Boehner said, “Absolutely not.”

“I believe that Congress needs to deal with this issue,” he said. “Our committees are continuing to do their work. There are a lot of private conversations that are underway to try to figure out how do we best move on a common-sense, step-by-step basis to address this very important issue.”

He also said he was “encouraged” when Mr. Obama earlier this week also signaled his willingness to accept a piece-by-piece approach.

“They’re suspicious of comprehensive bills, but you know what? If they want to chop that thing up into five pieces, as long as all five pieces get done, I don’t care what it looks like, as long as it’s actually delivering on those core values that we talk about,” Mr. Obama said at The Wall Street Journal’s CEO Council Annual Meeting.

Mr. Obama said he thinks many House Republicans want to give illegal immigrants a pathway to citizenship, which he said would involve paying fines and, for many, waiting more than a decade before they could obtain a green card.

Beneath this week’s apparent agreement, however, lurk several hurdles, including whether House Republicans insist that border security come before any legalization bill passes. Democrats have considered that a non-starter.

Mr. Boehner said Thursday that it’s the American people who are — and should be — skeptical of comprehensive bills.

“The only way to make sure immigration reform works this time is to address these complicated issues one step at a time,” he said.

⦁ Tom Howell Jr. contributed to this report.

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