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Still, Thomas S. Winkowski, a top official at U.S. Immigration and Customs Enforcement, said ICE agents have seen some signs of early successes in deterrence by opening more detention facilities to hold families who have come across the border. In the past, almost all of those families from Central America were released in the often futile hope they would return for deportation hearings.

“We’re already seeing people saying, ‘I didn’t realize I was going to detention; I thought I was going to be released,’” Mr. Winkowski said.

In his remarks Wednesday, Mr. Obama said Congress should change a 2008 law that says children from countries other than Mexico and Canada must be turned over to social workers rather than quickly deported, which is what happens to those from contiguous countries. Mr. Obama said he wants “flexibility” so he can decide in which cases deportations should be speeded up.

And there appears to be a bipartisan consensus developing around that change — though some immigrant-rights groups are pushing back.

“The vast majority of these children have relatives in the U.S., and it’s the humane thing to do to allow them to stay with loved ones, not deport them,” said Leoncio Velasquez, president of Hondurenos Unidos de Los Angeles.

Reversing course

The surge has forced several policy and rhetorical reversals from Mr. Obama. For years, he had said the border was finally secured — but his push for additional border security in his new spending package amounts to an admission that wasn’t true.

And while he said violence in Central America is driving children and families from those countries, he acknowledged that their understanding of U.S. immigration policy — even if it’s incorrect — is drawing them here.

On Wednesday the president said the surge is not a short-term problem, but he sought to push responsibility onto Congress, saying he’s taken all the steps he can on his own and that it’s up to lawmakers to pass the spending bill to boost resources.

But many on Capitol Hill say it is the president’s own slow response and his previous nondeportation policies that have made the problem worse.

And the way agencies have responded to Congress — often with tight controls over information — has not helped.

HHS officials turned away one Oklahoma congressman who made a surprise visit to a holding facility in his state, prompting a protest from Sen. Tom Coburn, Oklahoma Republican. He said members of Congress should always be allowed to visit — especially when they show up unannounced — as a key part of their ability to conduct oversight.

Mr. McCain, meanwhile, was furious at rules that prohibited him from taking photos of a Homeland Security holding facility in Nogales, Arizona, or speaking to the immigrants being held there.

U.S. Customs and Border Protection Commissioner Gil Kerlikowske said the rules were in place to protect the privacy rights of the illegal immigrant children, but Mr. McCain said that wasn’t going to cut it with him.

“I want it fixed, and I want it fixed immediately, understand?” the senator said. “If a member of Congress can’t visit a facility within his own state — the people of Arizona elected me, and I’m not supposed to even carry a cellphone with me? Mister, you have overstepped your responsibilities and your authorities.”