President Barack Obama invited Texas Gov. Rick Perry to a roundtable discussion in Dallas tomorrow on the surge of unaccompanied, undocumented children crossing the U.S.-Mexican border.

"The president would welcome a meeting with you while he is in Texas," senior adviser Valerie Jarrett wrote Perry on Monday on the president's behalf. Obama will be in Texas for political fundraisers in Dallas and Austin.

Perry earlier dismissed an invitation to meet Obama on the tarmac in Austin, the state capital. He instead suggested a private meeting on the surge of children and adults illegally crossing the U.S. border, according to the Austin American-Statesman newspaper.

Obama will be asking Congress on Tuesday to approve about $2 billion to contain the surge of illegal Central American migrants entering the U.S. at the South Texas border.

More than 52,000 unaccompanied children arrived at the U.S.-Mexico border from Oct. 1 through June 15, about double the total in a similar period a year earlier, U.S. Customs and Border Protection reported. Most of the children are smuggled through Central America and Mexico, according to the White House.

Perry, a potential Republican candidate for president in 2016, has said he's warned the Obama administration for more than two years about unaccompanied children showing up at the U.S.-Mexico border.

"I don't believe he particularly cares whether or not the border of the United States is secure," Perry said Sunday on ABC's "This Week."

Jarrett invited Perry to a meeting in Dallas with faith leaders and local elected officials to discuss the border crisis. Perry hasn't yet responded to the invitation to join Obama in the Dallas discussion, spokesman Eric Schultz said.

In asking Congress for more than $2 billion, the administration is seeking to bolster efforts to contain the influx of unaccompanied, undocumented children crossing the border.

The administration plans to increase detainment and boost court capacity to speed decisions, expand law enforcement and prosecution of criminal networks, seek greater cooperation with Central American countries and increase the amount of housing, care and transportation while migrant cases are judged.

Most of the children crossing the border unaccompanied by a parent probably won't qualify for humanitarian relief and will be deported, Obama's spokesman said.

"It's unlikely that most of the kids who go through this process will qualify for humanitarian relief, which is to say that most of them will not have a legal basis," press secretary Josh Earnest said. He said courts, in most cases, would rule that migrants don't have a legal reason to be in the U.S.

In Congress, key Republican senators said they'll seek spending cuts to pay for an emergency request, while the administration is seeking to treat the request as "emergency" spending, which would add to the deficit.

"Absolutely it ought to be offset," the Senate Appropriations Committee's ranking Republican, Richard Shelby of Alabama, said yesterday. "It ought to be offset from Obamacare."