In a news conference on Sunday evening, Mr. Trump expressed his confidence in the federal response, including his administration’s relationship with governors and the capacity for testing.

Mr. Trump said the administration was preparing to use the Defense Production Act to compel one U.S. facility to increase production of test swabs by over 20 million per month. The announcement came after he defended his response to the accusations that there was an insufficient amount of testing to justify reopening the economy any time soon.

“You’ll have so many swabs you won’t know what to do with them,” Mr. Trump said.

Officials at every level have faced increasingly competing pressures, balancing maintaining stay-at-home orders against the exasperation and economic toll they are producing. On Saturday and Sunday, modest protests took place in several cities across the country, where demonstrators flouted social distancing rules as they demanded that restrictions be relaxed.

Yet there was also a widespread sense that much of the public understood the governors’ concerns and shared them. Nearly 60 percent of American voters said they were worried that measures would be relaxed too soon, causing deaths to rise, according to a new poll from NBC News and The Wall Street Journal.

Officials in various states said they had started staging plans for reopening their economies and were working in concert with neighboring states in determining when to lift restrictions.

In South Carolina, Gov. Henry McMaster said that he had spoken with the governors of other southeastern states, including Florida and Tennessee. “Told them South Carolina was ready,” Mr. McMaster, a Republican, said on Twitter on Saturday.

On Sunday, governors from across the Northeast, including New York, New Jersey, Connecticut and Pennsylvania, said they were creating a regional council focused on restoring the economy and addressing unemployment.