"What's happened is these children are placed with family members in the United States and given a notice to appear for a later court hearing. Some have called this a notice to disappear, not a notice to appear," as most don't show up, he said.
Another Republican, Sen. Ron Johnson of Wisconsin, said the United States should immediately fly the children back to their home countries. It would cost less and signal U.S. intolerance for those who enter the country illegally, he said.
Obama's emergency funding request seeks $1.6 billion to bolster customs and border efforts as well as crack down on smugglers.
Another $300 million would go to help Mexico and Central American governments counter claims by smugglers to desperate parents that U.S. officials won't send their children back.
"While we intend to do the right thing by these children, their parents need to know that this is an incredibly dangerous situation and it is unlikely that their children will be able to stay," Obama said.
The request also includes $1.8 billion to provide care for unaccompanied children crossing the border.
The request is about 10% of the $30 billion in proposed border security funding included in the immigration reform bill passed by the Senate but stalled in the Republican-controlled House.
Perry said lawmakers aren't going to back Obama's funding request if the government doesn't act to stop policies they see as encouraging children to show up at the border.
He told Obama in a meeting that he wants 1,000 National Guard troops deployed to help secure the border, an option Obama seemed less than enthusiastic about.
"What I told him was we're happy to consider how we could deploy National Guard down there, but that's a temporary solution," Obama said of his discussion with Perry. "That's not a permanent solution. And so why wouldn't we go ahead and pass the permanent solution, or at least a longer-term solution?"
It was unclear how much cooperation Obama will get from congressional Republicans on his funding request. A spokesman for House Speaker John Boehner said the request would be considered.
Cornyn and Cuellar plan to introduce legislation Thursday to repeal the 2008 deportation hearing law. The bill, Cuellar said, will include protections for children who come to the United States because of concerns about drug or sex trafficking.
Democrats want to keep the law intact to ensure that children who deserve asylum receive a full hearing.