Health and Human Services Secretary Sylvia Matthews Burwell met privately with dozens of governors Sunday as the Obama administration tried to get support from the leaders of states that will host thousands of the Central American children who have crossed the Mexican border on their own since Oct. 1.

Governors of both parties expressed concerns about the cost to states, including providing public education for the children, according to those who attended the meeting. Burwell left the meeting through a side door without talking to reporters.

“Our citizens already feel burdened by all kinds of challenges. They don’t want to see another burden come into their state,” said Colorado Gov. John Hickenlooper, a Democrat. “However we deal with the humanitarian aspects of this, we’ve got to do it in the most cost-effective way possible.”

Wisconsin Gov. Scott Walker, New Jersey Gov. Chris Christie and Iowa Gov. Terry Branstad were among the most vocal Republican critics. They seized on the administration’s plans to place the children with friends or family members without checking on their immigration status.

Under current law, immigrant children from countries that don’t border the United States and who cross into this country by themselves are turned over to HHS within 72 hours. From there, they often are reunited with parents or placed with other relatives already living in the country, while they wait for an immigration court to decide their future. The court process can take years.

Neither Burwell’s agency nor immigration officials check the immigration status of relatives who take custody of the immigrant children.

Since Oct. 1 more than 57,000 children have crossed the border alone. Most are from Honduras, El Salvador or Guatemala.

“We want to make sure they’re placed in a safe and supportive home or placement, but also, it should be somebody that is legal and somebody that will be responsible to see that they show up for the hearing,” Branstad said.

Amid the debate of what is causing the ongoing crush of child immigrants and how the government can stem the flow, two key lawmakers said President Barack Obama can take administrative action to relieve much of the crisis without waiting for what is likely to be a contentious and lengthy Congressional battle.

At issue is a provision in a 2008 human trafficking law that puts the fate of these immigrants in the hands of immigration judges. The Obama administration has expressed some interest in asking Congress to change the law to give the administration more leeway in dealing with the crisis.

But Rep. Mike Rogers, a Michigan Republican and chairman of the House Intelligence Committee, said Sunday that wholesale changes by Congress may not be necessary and that Obama has the authority to return the children to their native countries.

“We can safely get them home,” Rogers said on NBC’s “Meet the Press.” “And that’s where the president needs to start. So he needs to re-engage, get folks who are doing administrative work on the border. They need to make sure they send a very clear signal.”

Democratic Sen. Dianne Feinstein of California, the author of the provision in the human trafficking law, said a change in regulations, not the law, could speed the children’s return.