(CNN) -

Unlike some of his fellow Republicans, Texas Gov. Rick Perry declined to place direct blame on President Barack Obama for the recent influx of children illegally crossing the border from Mexico into the United States.

Perry was given multiple opportunities at a congressional hearing Thursday in Texas to fault the Obama administration for what the White House and Republicans are calling a "humanitarian crisis."

And while Perry criticized the administration for failing to secure the border in general, he said he was looking for "solutions," not to find fault in the latest series of events involving minors.

"I'm tired of pointing fingers and blaming people," he said at the hearing, which was hosted by the House Homeland Security Committee and held near the border in McAllen, Texas. "I hope what we can do is come up with some solutions here."

However, he urged the President to send a clear signal that people cannot cross the border without consequence or else there "will be a deluge."

Perry, who's considering a second run for president, also called on members of Congress to work with Obama.

"The President has bypassed Congress--and maybe he could and should, I don't know, I'm not going to wade in--but this is the one time that I really think the President does need all of you, Democrats and Republicans alike, to be working with you and reaching out," he said.

"I hope that all of us will share with the President that we're willing to sit down with him and work with him," he continued.

The outgoing governor was also not shy in calling on the President to visit the border and do more to secure the U.S.-Mexico line. And he pushed back against those who argue that the children who crossed the border illegally should be allowed to stay.

"Nobody is doing any of these children the slightest favor by delaying a rapid return to their countries of origin," he said. "Allowing them to remain here will only encourage the next group of individuals to undertake this very, very dangerous and life threatening journey."

The Obama administration has accused syndicates in Latin America of waging a deliberate campaign of misinformation that has spurred people in poor Central American countries and Mexico to risk their lives to cross the border illegally.

While he said he "readily" welcomes the $2 billion in emergency funds for the border that Obama announced earlier this week, the governor asked that Texas be reimbursed for more than $500 million that the state has spent on border security over the past decade.

Texas Attorney General Greg Abbott, this year's GOP nominee for governor, told the conservative news outlet Breitbart that he's thinking about suing the federal government to get back money the state has spent on the recent crisis involving children. Authorities estimate 60,000 to 80,000 children without parents will cross the border this year.

White House Press Secretary Josh Earnest said Thursday "it's hard to take seriously Governor Perry's concerns" when his Republican colleagues in the U.S. House have failed to pass bipartisan immigration reform like the Senate did last year.

"I think the most effective way that Gov. Perry can help, if that's what he wants to do, would be to pick up the phone and call the Republican members of the House of Representatives that represent the state of Texas and tell them to support the bipartisan proposal to reform our immigration system that passed through the Senate," he said.

Last month, the Obama administration unveiled a plan to spend almost $100 million in aid to Guatemala, Honduras and El Salvador to help reintegrate the undocumented migrants whom the United States will deport, and to help keep them in their home countries.

The administration also will set aside $161.5 million this year for the Central American Regional Security Initiative programs in an effort to "help stem migration flows as well as address the root cause of the migration," the White House said.