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The companies, however, said they had been ordered by the government not to speak to the press about their work, and an HHS spokesman would not say why.

Kenneth J. Wolfe, deputy director of the Office of Public Affairs, said the details of the work are included in the contracts with the companies, but those contracts had to be obtained under open-records requests. Those requests can take months to fulfill.

In the meantime, federal spending databases give a glimpse of past work and the skyrocketing costs already incurred this year.

HHS spending through the first nine months of the fiscal year totaled $366.7 million, marking a huge increase over just a few years ago, when the government spent $136.2 million for all of 2011.

From 2000 through 2014, one company, Southwest Key Programs Inc. has collected more than $328 million in taxpayer contracts, while Baptist Child & Family Services has taken another $232.9 million.

One major challenge has been transferring the 10,000 or so children caught at the border each month to other locations to be processed, then sending them to the HHS-sponsored facilities to be held. The White House spending request includes $116 million for transportation costs.

Before the children are turned over to HHS, they are held by Homeland Security authorities — as are the Central American families also surging across the border.

The Homeland Security Department has recorded tens of thousands of dollars in expenses for clothing, shoes, toiletries, translation costs, office equipment and “housekeeping,” according to spending records.

The children’s games were purchased for use at the detention facility being set up at the Federal Law Enforcement Training Center in Artesia, New Mexico. The $584 tab, racked up at Staples, covered four Monopoly games, six ring tosses, 12 beanbag tosses, 12 tic-tac-toe games, five packs of playing cards and eight checkerboards.

The spending records also show the difficulties of finding places to house some of the children.

One company signed an agreement to run a facility in Lawrenceville, Virginia, providing “shelter and mass care support” for unaccompanied minors. That plan was canceled after local residents protested.