The remarks imposed an urgent timeline on the guidance the governor has been giving for weeks — that if New York did not get a major infusion of the potentially lifesaving machines, and quickly, the number of virus-related deaths in the state would spike drastically.

“If a person comes in and needs a ventilator and you don’t have a ventilator, the person dies,” Mr. Cuomo said at his daily briefing in Albany. “That’s the blunt equation here. And right now we have a burn rate that would suggest we have about six days in the stockpile.”

The comments came as doctors in New York City, where hospitals’ supplies are dwindling amid a flood of virus patients, cautioned that medical workers might soon need to make difficult choices about rationing care.

Across the United States, hospitals and public health officials have been working on plans for what might happen if the number of virus patients were to exceed the available space in intensive care units.