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Still, the CBO had glowing news for the federal budget, saying that increased taxes from legalized immigrants and future workers will decrease deficits by $197 billion over the next 10 years and nearly $900 billion over two decades.

Even as the Senate debated amendments, the House Judiciary Committee was moving ahead with a bill that would impose criminal penalties on illegal immigrants, give state and local police more authority to enforce immigration laws, and push federal authorities to act.

Democrats objected, saying Republicans were repeating the mistakes of 2005 and 2006, when House Republicans passed an enforcement-only bill, sparking mass protests that some analysts said was the beginning of the political wave that struck in the 2012 elections.

“We’ve tried this before and it has failed us before,” said Rep. John Conyers Jr., the ranking Democrat on the committee. “It is so extreme and heinous that this committee can do nothing but reject this bill, consider it a nonstarter, because the bill cannot be fixed.”

The enforcement bill is the first in a series of piecemeal bills on which the Judiciary Committee is working. That stands in contrast to the 1,075-page bill the Senate is debating.

But another bipartisan group of lawmakers is working on a broad bill in the House, and Mr. Boehner hasn’t decided which approach he will take when it’s time to bring a bill to his chamber’s floor.

Mr. Boehner said he will gather House Republicans on July 10 to talk about how to proceed. He acknowledged that if he gets this issue wrong, his colleagues could oust him as speaker.

Some Republicans fear Mr. Boehner will push a tough bill through the House but then allow a weaker bill to come back after negotiations with the Senate.

Asked whether he would apply his same majority-of-Republicans rule to any eventual conference report negotiated with the Senate, Mr. Boehner demurred.

“We’ll see when we get there,” he said.