While 57,000 immigrant children have entered the U.S. since October, another 30,000 are expected in the next 2½ months.

Secure the border

The two sides are also sharply at odds over the issue of border security. Republicans say too little money is in the President's emergency request to protect the border. Of the $3.7 billion request, $433 million would be for Customs and Border Protection, the agency in charge of border security.

"The way to deal with it is to secure the border first," Rep. Marsha Blackburn, R-Tennessee, said on CNN's "State of the Union."

Texas Gov. Rick Perry is leading the Republican charge for additional resources at the border.

"If you have a patient who is bleeding profusely, the first thing you have to do is stop the bleeding, and that's the reason we have been so adamant about securing the border," Perry said on "Face the Nation."

He is calling on Obama to send 1,000 National Guard troops to the border that could "show the force."

Those troops would be temporary, until an additional 3,000 Border Patrol agents would be permanently placed.

"Gov. Perry's just wrong," is Gutierrez's reaction. "The border is secure; the fact is, the children are handing themselves over to the Border Patrol agents."

McCain said the U.S. needs to spend "about $6 billion" to secure the borders.

Since 2002, spending on border security has more than doubled. The budget for Customs and Border Protection has ballooned from $5 billion to $12.4 billion in just over a decade, with steady increases each year, according to Homeland Security Department budget documents.

The number of Border Patrol agents have also more than doubled. More than 18,611 agents patrolled the southern border in 2013, which comes to about 9.7 agents per mile, according to Customs and Border Protection.

But on "Fox News Sunday," Perry insisted that more money needs to be spent.

Obama needs to be "realistic about the problem and how you deal with the problem -- and it is a border security issue."

"You can keep throwing money and talk about enforcement, enforcement, enforcement, but you've got to put money also into your judicial system, and you've got to put money in a comprehensive program that deals with the issue," Gutierrez responded.

Rep. Mike Rogers, R-Michigan, succinctly framed the dynamics of the current debate on NBC's "Meet the Press."

It "certainly appears most of the parties have gone to their mutual corners," he said. "We've got to get past that."