The back and forth came as federal officials have projected a particularly bad week ahead, as many Americans prepared to observe Passover and Easter without traditional services or family gatherings. Three major metropolitan areas — New York, Detroit and New Orleans — have seen death rates rise rapidly, and other states are anticipating a peak in cases later this month.

In New York, where the death toll has climbed to over 4,000 people, with more than 122,000 confirmed cases, Gov. Andrew M. Cuomo said on Sunday that there were indications of “slight plateauing.” But he noted that it was too soon to say whether the changes in the numbers were indicative of a trend.

In Louisiana, Gov. John Bel Edwards said his state could run out of ventilators as soon as Thursday. And Mr. Cuomo described the logistical struggle in New York to get hospitals the equipment they need.

“I can’t say to a hospital, ‘I will send you all the supplies you need, I will send you all the ventilators you need,’” he said in a briefing on Sunday. “We don’t have them. It’s not an exercise. It’s not a drill. It’s just a statement of reality. You’re going to have to shift and deploy resources to different locations based on the need of that location.”

The increasingly dire need for ventilators and protective equipment has touched off a race between states, cities and hospitals to buy up the equipment. In a conference call with the nation’s governors last month, Mr. Trump told the states that the federal government would be backing them, but that they should try to procure their medical supplies themselves.

“We’ve been outbid by another state after we had the order confirmed, so yes, that has been challenging for us,” Gov. Asa Hutchinson of Arkansas, a Republican, said on “Meet the Press.” “But we recognize that the federal government has said, ‘We’re your backstop. You’ve got to get out there and compete.’ And it literally is a global jungle that we’re competing in now.”

Governors have aired their frustration as they called on the Trump administration to do more, like using the federal Defense Production Act more aggressively to ramp up manufacturing of the necessary equipment or putting in effect a national stay-at-home order. Many governors have been tempered in their criticisms, acknowledging the need for a solid working relationship with the president as the crisis continued to grow.