The program had to start borrowing from the Treasury in 2005, when it needed $17.5 billion in fresh cash to pay big flood claims filed after Hurricanes Katrina, Rita and Wilma. In 2012, it borrowed another $6.25 billion to pay claims filed after Hurricane Sandy. As now structured, the program has no way to repay the Treasury, and is carrying the debt forward on its books, year after year.

Private insurance companies get the money they need to pay claims by charging policy holders premiums and building up carefully calculated reserves. The National Flood Insurance Program charges premiums, too, but most experts agree they have fallen far behind the rising cost of repairing homes and businesses after catastrophic flooding.

As of this week, the program had just $5.8 billion available to pay claims, the FEMA spokeswoman said. More than 88,000 claims have been submitted from Hurricane Harvey, and more than 27,000 from Irma.

Asking lawmakers to take action on the program’s debt, Mr. Mulvaney wrote that the hurricanes had “inflicted projected losses of $16 billion,” and that without fresh money, the program will have exhausted its resources and be unable to pay claims “by the latter part of this month.”

Fiscal conservatives in Congress, especially those from states without much flooding, may object to the debt forgiveness because it means the taxpayers they represent will not be repaid for helping to pay flood claims in Louisiana, Texas, New York and New Jersey.

To make the taxpayer losses more palatable, the White House also proposed on Wednesday to raise flood insurance premiums, at least for homeowners who can afford them, and to make changes that would allow insurance companies to underwrite flood insurance themselves.

Currently, insurance companies can sell flood insurance, but they do not pay the claims with their own money. Instead, they collect fees for marketing and administering the policies and pass the risk on to the federal government.