While a national stockpile of ventilators exists, it is unclear how that will be doled out. In a conference call on Monday with a group of governors, including Gov. Andrew M. Cuomo of New York, President Trump said that states should not count on the federal government for more ventilators and other equipment. “We will be backing you, but try getting it yourselves,” he said, according to a recording of the call.

Most of the New York’s major health systems have declined to provide details about their ventilators or what they will do if they run out. Some hospital administrators have said they believe new measures, such as closures of schools and restaurants, can slow the spread of the virus and keep the number of critically ill people below levels that could overwhelm hospitals.

But officials have repeatedly said that a ventilator shortage is one of their top concerns.

“These ventilators are expensive to begin with, and they are scarce,” Mr. Cuomo said on Sunday. “You’re going to be thousands short. Thousands.”

In interviews, ventilator manufacturers warned that, amid the pandemic, they do not have the ability to provide New York more ventilators — which can cost at least $25,000 apiece and require significant training to use.

“I’m nervous,” said Dr. Akshay Ganju, an emergency room physician in Westchester County. “I’m steeling myself for the possibility that at some point I’m going to have to look a daughter in the eye and tell her that I don’t have a ventilator for her father.”