On Tuesday, officials in Michigan said the state had received three shipments of supplies from the federal government, including about 1 million masks and hundreds of thousands of gloves, allocated by population size. They said they had also requested additional supplies from the federal government: Of thousands of ventilators requested, they had received about 400.

All of the supplies sent to Michigan had been working and usable, a state representative said. In California, officials said 170 ventilators sent to Los Angeles had been unusable and were sent to a Silicon Valley company for repair.

Mr. Trump struck a cooperative tone at the White House in a news conference on Tuesday as he detailed his calls with governors to a group of reporters. He said that he had spoken to Ms. Whitmer and that a field hospital with 250 beds would be established in Michigan.

The need for more supplies has become a central issue as the coronavirus has spread, foretelling months of crisis. One model, created by scientists at the University of Washington’s Institute for Health Metrics and Evaluation, predicts that deaths from the virus in the United States will rise rapidly during the month of April, from about 4,000 to almost 60,000, even with the many restrictions on movement now in place. The study suggests that the pace of deaths will eventually slow down, reaching a total of about 84,000 by the beginning of August.

Mr. Trump recently extended federal social distancing guidelines at least through the end of April.

White House officials have struggled with the idea of federal mandates, preferring to take a traditionally conservative small-government approach to the crisis. Officials have at times discussed the idea of a national curfew or similarly stringent measures — only to have those dismissed by higher-ranking players.

That has left governors to make their own decisions about whether to take the significant step of ordering residents to stay home, creating a split across the United States, with more than 30 states issuing some type of statewide stay-at-home instruction.

Gov. Ron DeSantis of Florida, who has been criticized for refusing to set more statewide mandates as the number of confirmed cases of the coronavirus there has soared over 6,700, said on Tuesday that one reason he had not was because the White House’s task force had not recommended it.