The Biden administration’s chief medical adviser said he didn’t believe the U.S. would return to lockdowns but warned that “things are going to get worse” as a more contagious variant of the coronavirus has led to a surge of new cases.

“We are looking, not I believe, to lockdown but we are looking to some pain and suffering in the future because we are seeing the cases go up,” Dr. Anthony Fauci, director of the National Institute of Allergy and Infectious Diseases, said Sunday on ABC’s “This Week.” He added, “The solution to this is, get vaccinated.”

The latest statistics from the U.S. Centers for Disease Control and Prevention show a 64.1% increase in Covid cases over the week ended July 30 compared with the previous week, or an average of 66,606 cases a day. The CDC reported a current seven-day average of 6,071 new admissions of hospital patients with Covid-19, a 44% increase over the average for the week of July 16-22. CDC Director Rochelle Walensky previously has said that more than 97% of Covid patients entering the hospital nationwide were unvaccinated.

As of July 26, the CDC had reported fewer than 1,000 Covid-related deaths among vaccinated individuals. More than 164 million people in the U.S. have been fully vaccinated.

Dr. Fauci and other top medical officials pleaded on several Sunday talk shows for Americans to follow the newest government guidelines—that everyone in areas with high Covid-19 infection rates wear a mask, regardless of vaccination status. They also urged the nearly 100 million eligible Americans who hadn’t received a vaccine to get one.

While Covid-19 vaccinations are highly effective at preventing hospitalizations or death from the virus, they’re not foolproof in preventing infection. This poses problems for events like the Olympics and raises broader questions about immunity in the long term. Photo: David Crigger/Bristol Herald Courier/Associated Press The Wall Street Journal Interactive Edition

Appearing Sunday on CNN’s “State of the Union,” Francis S. Collins, director of the National Institutes of Health, said the new guidelines are “mostly about protecting the unvaccinated.”

Vaccinated people who get breakthrough infections of the Delta variant of Covid-19 can spread it to other people, even if they have few or no symptoms, both doctors said. The Delta variant of the virus stays in the nasal cavity of the vaccinated at the same levels of the unvaccinated, they said, which is why it is more contagious than earlier forms of the virus.

The new government guidelines were based, in part, on a study released last week by the CDC of Barnstable County, Mass., where among the 469 cases linked to an outbreak in the county, nearly 75% were fully vaccinated. The outbreak helped prompt the CDC to urge some Americans to start wearing masks again, the agency said, because it demonstrated that vaccinated people infected with the Delta variant might be as contagious as those who are unvaccinated.

“The fact is, if you get infected, even if you are without symptoms, you very well may infect another person who may be vulnerable,” Dr. Fauci said on ABC. “So, in essence, you are encroaching on their individual rights.”

The transmission rate of the Delta variant, coupled with the number of unvaccinated Americans, has led to a surge of new cases, the doctors said.

“That’s where the real serious risks of illness are,” Dr. Collins said.

While the number of cases in the U.S. had climbed, the number of Americans getting vaccinated also surged, particularly in states like Louisiana, Arkansas and Mississippi, where the rates of vaccination were lower, the officials said. Roughly 60% of eligible Americans are fully vaccinated.

Write to Nancy A. Youssef at nancy.youssef@wsj.com