On ABC’s “This Week,” Adm. Brett Giroir, an official with Health and Human Services, acknowledged that with “more cases, more hospitalizations,” the expectation was for “deaths to go up” over the next several weeks. “It’s really essential to wear masks,” he said, adding: “We have to have like 90 percent of people wearing the masks in public in the hot spot areas. If we don’t have that we will not get control of the virus.”

The host, George Stephanopoulos, asked about suggestions by Mr. Trump — who after months of refusing, donned a mask on Saturday during a visit to Walter Reed National Military Medical Center — that there could be some harm in wearing face coverings.

“There’s no downside to wearing a mask,” Admiral Giroir said. “I’m a pediatric I.C.U. physician. I wore a mask 10 hours a day for many, many years.”

Asked if states with stark increases in cases, like Florida, South Carolina, Arizona, Texas and Georgia, should consider more stringent measures, Admiral Giroir said that closing bars and limiting the number of patrons allowed in restaurants are “two measures that really do need to be done.”

The admiral, who has been in charge of testing, also told NBC’s “Meet the Press” that rates of people testing positive were “leveling off.” However, the Covid Tracking Project at The Atlantic shows positivity rates leveling only in the Northeast, with rates rising in the South, West and Midwest.

Surgeon General Jerome Adams said people needed to “understand the importance of wearing face coverings and good hand hygiene and staying home when they can.”

Dr. Adams wore a mask during his entire interview on the CBS program “Face the Nation” even though he was being interviewed remotely from Indiana. He said measures like wearing face coverings were “critically important.”