Updated

The Trump administration’s plan to ship illegal immigrants who enter the U.S. through Mexico back across the border regardless of their homeland could be one of many thorny issues Secretary of State Rex Tillerson will face on his first official trip to Mexico City on Wednesday.

A pair of DHS enforcement memos released Tuesday effectively carried out the executive actions on immigration that President Trump signed last month. They covered everything from the planning and construction of a U.S.-Mexico border wall to stepped-up deportations to the hiring of more enforcement agents.

But in one of the biggest shifts, the DHS called for using a long-standing but obscure U.S. law to send some illegal immigrants back to Mexico even if they are from other countries.

Previous U.S. policy called for only Mexican citizens to be sent to Mexico. Migrants known as "OTMs" — Other Than Mexicans — got flown back to their homelands.

Under the updated policy guidance, the U.S. would return certain illegal border crossers “to the foreign contiguous territory from which they arrived.” According to DHS, doing so “saves the Department’s detention and adjudication resources for other priority aliens.”

But some advocates and officials in Mexico – where President Trump’s immigration policies have long been unpopular – already are saying they’re not prepared to take those migrants back.

“Not in any way, shape or form," the Rev. Patrick Murphy, a priest who runs the Casa del Migrante shelter in the border city of Tijuana, told The Associated Press.

It's unclear whether the United States has the authority to force Mexico to accept third-country nationals.

"I hope Mexico has the courage to say no to this," Murphy said.

Victor Clark, director of Tijuana's Binational Center for Human Rights, said Mexico can simply refuse to accept non-Mexican deportees. "They come through one by one, and when the Mexican immigration agent sees a person who isn't Mexican, he tells the ICE agent, 'I can't accept this person, he's not Mexican,' and they return him to the United States."

The new policy could result in new deportee and refugee camps along the border, raising the question of which government should pay for new facilities.

Mexican officials repeatedly have rejected Trump’s demand that they foot the bill for his ordered southern border wall. For Tillerson, the deportation change is yet another issue fueling tensions between the U.S. and Mexico that he likely will confront on his visit this week – along with questions about the wall, U.S.-Mexico trade and more.

Mexico's government did not formally react to the DHS policy statements.

But in a hearing with Mexican senators, Mexico's new ambassador to the United States, Geronimo Gutierrez, said, "Obviously, they are a cause for concern for the foreign relations department, for the Mexican government, and for all Mexicans."

Gutierrez, though, praised the Trump administration's release of the policies before Tillerson’s visit, calling that "a position that is much more straightforward and honorable, to make these positions known beforehand ... so they can be discussed."

There are precedents in Mexico for refugee camps.

In the 1980s and 1990s, Mexico took in about 46,000 Guatemalans fleeing civil war. With help from the United Nations, camps were set up in the southern states of Chiapas, Campeche and Quintana Roo. When peace accords were signed in Guatemala in the mid-1990s, almost 43,000 refugees and their children went home, but more than 30,000 Guatemalans and their children born in Mexico decided to stay.

The same thing could happen with any migrants housed in Mexico.

The Associated Press contributed to this report.