Migrant children, many of whom have been separated from their parents under the "zero tolerance" policy by the Trump administration, are being housed in tents next to the Mexican border in Tornillo, Texas, U.S. June 18, 2018.

The U.S. government said it still had 2,053 children in its custody who were separated from their parents under President Donald Trump's "zero tolerance" immigration policy, and set out its most detailed plans yet on how it would reunite families.

The Department of Homeland Security (DHS) said late on Saturday it had a "well coordinated" process in place — in the face of criticism from lawyers for parents and children who have said they have seen little evidence of an organized system.

A total of 522 children had already been reunited with parents, the agency added in a factsheet published three days after Trump ended his policy of separating families on the U.S.-Mexico border, after images of youngsters in cages triggered outrage at home and abroad.

"The United States government knows the location of all children in its custody and is working to reunite them with their families," the DHS said.

The new details came after more than two months of confusion how detained migrant parents, who are shuttled from facility to facility run by different government agencies, would ever reunite with their children, who are sent to shelters and foster homes scattered across the country.

The factsheet said the Trump administration has a process for how parents would be reunited with their children "for the purposes of removal," or deportation.

Deportation proceedings could take months to complete, and the factsheet did not say whether parents and children would be reunited in the intervening time. DHS officials did not immediately respond to questions about the process explained in the fact sheet.