"The speaker still supports deploying the National Guard to provide humanitarian support in the affected areas - which this proposal does not address," said Michael Steel.
On the Senate side, Republicans argued for changing a 2008 law signed by GOP President George W. Bush that requires deportation hearings before sending back children from non-bordering countries.
They blame the process for a backlog created by the surge in unaccompanied minors from Central America entering the United States illegally from Mexico.
"I don't think we can solve the problem unless we revisit" the law, said conservative Republican Sen. John Cornyn of Texas.
Democrats, however, want to keep the law intact to ensure that any children who deserve asylum get due process in the form of a full hearing.
"I'm not inclined to support any policy change that ultimately undermines existing law and would violate the right of someone who is actually a legitimate refugee," said Sen. Robert Menendez of New Jersey, a Democrat of Cuban descent who chairs the Senate Foreign Relations Committee. "Each child should be able to make their case as to why they qualify. Many will be sent back but others will qualify."
No border visit
Obama has come under criticism from Republicans and some Democrats for not planning to visit the border area during his Texas trip, which will include a Democratic Party fundraising event.
Former Arkansas Gov. Mike Huckabee, a possible GOP presidential nominee in 2016, called the situation similar to the much-disparaged federal response to Hurricane Katrina by the Bush administration.
"For him to go to Texas and spend two days shaking down donors and never even getting near the border mess he helped create would be like flying into New Orleans in the highest waters of Katrina to eat Creole cooking, but never getting near the 9th Ward, the Superdome, or the Convention Center where thousands languished in squalor," Huckabee said.
Meanwhile, Democratic Rep. Henry Cuellar of Texas questioned if the border problem would harm the President in the same way the Katrina response tarnished his predecessor, telling Fox News: "I hope this does not become Obama's 'Katrina moment.'"
The White House officials on Tuesday's conference call repeatedly emphasized that Obama regarded the immigration crisis as "an urgent humanitarian situation."
Obama's trip to Texas set off a political squabble, with Perry refusing to greet the President at the airport and instead calling for a meeting to discuss the immigration crisis.
In response, Obama invited Perry to a meeting in Dallas with faith leaders and local officials, according to a letter written by senior adviser Valerie Jarrett.
Perry, who is seeking to re-establish his national credentials after a disastrous bid for the GOP presidential nomination in 2012, said Tuesday he looked forward to meeting with Obama.
Previously, Perry has said he "readily" welcomes any federal emergency funds, but also has asked that Texas be reimbursed for more than $500 million the state has spent on border security over the past decade.
Secretary of Homeland Security Jeh Johnson said Sunday the administration will take steps to fix the nation's broken immigration system, even without the help of Congress.
That's the same message Obama has given in response to the refusal by House Republicans to take up a Senate-passed immigration reform bill.