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But there are signs the Senate bill may not be as popular in the House as Mr. Reid and Mr. Schumer think.

While the bill won unanimous support of Senate Democrats, some House Democrats have objected to the massive boost in Border Patrol agents along the U.S.-Mexico border.

And in a letter on Wednesday, House Minority Leader Nancy Pelosi shied away from backing the Senate bill, instead saying it moved the country “one step closer” to a final agreement.

In a letter to Mr. Boehner, she urged the House to work on its own solution in a bipartisan way.