WASHINGTON — The Trump administration has warned Congress that the flow of unaccompanied migrant children at the southern border has increased even further than it anticipated just a few weeks ago and may require an additional $1.4 billion to provide housing and care.

In a letter to congressional leaders dated Friday, the administration said the extra money would be needed on top of $2.9 billion it had already requested to deal with the growing influx of migrant children. The immigration rate has already surpassed the projections used to justify that request, which was sent on May 1.

“Since then, the situation has continued to deteriorate and is exceeding the previous high-end estimate,” Russell T. Vought, the acting director of the White House Office of Management and Budget, wrote in the letter to lawmakers. The rising demands “will exhaust amounts currently available to fund services that involve the safety of human life, protection of property and the immediate welfare of individuals.”

The rising costs underscored that while the president in his public speeches and rallies focuses on the danger of criminals entering the country, the immediate crisis at the border is a humanitarian one — caring for the sick, housing children, holding families in temporary detention facilities and paying for extra asylum screeners.