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If the government shutdown continues into November, about 42 million low-income people could face severe disruptions to their food stamp benefits, the Agriculture Department warned in a letter to state agencies last week, saying that the federal government would have “insufficient funds.”

More than a dozen states have since warned that food stamp recipients may experience significant delays in obtaining benefits next month, see their aid reduced or not receive assistance at all.

The letter, a copy of which was obtained by The New York Times, said that the Agriculture Department’s Food and Nutrition Service, which operates the food stamp program, known as the Supplemental Nutrition Assistance Program, or SNAP, was exploring contingency plans. But it directed state agencies to pause sending vendors the electronic files typically used to load the benefits for November.

“We’re going to run out of money in two weeks,” Brooke L. Rollins, the agriculture secretary, told reporters at the White House on Thursday. “So you’re talking about millions and millions of vulnerable families, of hungry families that are not going to have access to these programs because of this shutdown.”

In a statement, a White House official said that Democrats “chose to shut down the government knowing that programs like SNAP would soon run out of funds.”

Such a disruption would be the first in recent decades. Benefits have remained available through every shutdown in the last 20 years, said Carolyn Vega, the associate director of policy analysis for Share Our Strength, a nonprofit that supports antipoverty programs.

“We are in uncharted territory,” she said.

Nearly 42 million people participated in the food stamp program in May, the month with the latest available data. They received an average of $188, at a total cost of about $8 billion.

Nutrition policy experts pointed to several other funding sources for the food stamp program, and noted that the Agriculture Department could also provide partial benefits for November. The agency could tap into a contingency fund of about $6 billion, but that would still fall short of covering full benefits for the month.

To fund a federal nutrition program for mothers and children known as WIC, the Trump administration used money collected from customs duties. It is possible that the Agriculture Department could use that same source, known as Section 32, but the account is largely used for school lunch and other child nutrition programs, and is unlikely to be sufficient to fund both food stamps and WIC.

The agency also has regulations on how to reduce benefits based on need and household size. For example, higher-income families could see their November benefits reduced by a larger percentage than lower-income ones.

The Agriculture Department could also turn to a specific interpretation of existing law to justify continuing to fund food stamps, said David A. Super, a law professor at Georgetown University. Under that theory, food stamps are an entitlement program, like Medicare, that is not subject to the annual appropriations process.

“The simplest approach for the U.S.D.A. would be to recognize that language in the Food and Nutrition Act makes SNAP an entitlement independent of appropriations, and continue paying benefits on the strength of that language,” Mr. Super said.

Several states, such as Illinois and New York, have already stated that they cannot provide funding from their own coffers. And at least one state is already warning that October benefits, too, may be affected. Minnesota’s Department of Children, Youth and Families said on Friday that it could not guarantee the availability of benefits before Nov. 1 for new enrollees.

Still, Ms. Vega cautioned against panicking, emphasizing that eligible people should still apply. She called for more clarity and a quick resolution from officials in Washington.

“While a delay is certainly better than not issuing November benefits at all, even that can be really significant to a family that is counting on that money and already has a tight budget,” she said.