WASHINGTON—Top U.S. public-health officials on Sunday voiced support for Covid-19 vaccination mandates imposed at the local level, while the head of a national teachers union also backed such a move in schools.

“Mandates at the local level need to be done,” Dr. Anthony Fauci, director of the National Institute of Allergy and Infectious Diseases, said on NBC’s “Meet the Press.”

Dr. Fauci said that once Covid-19 vaccines receive full approval from the Food and Drug Administration, a step that he said he hoped would come in a few weeks, “you’re going to see the empowerment of local enterprises giving mandates. That could be colleges, universities, places of business—a whole variety—and I strongly support that.”

His comments and similar ones Sunday from Francis Collins, director of the National Institutes of Health, come as more businesses, schools and local entities have instituted vaccine requirements. “I celebrate when I see businesses deciding that they’re going to mandate that for their employees,” Dr. Collins said on ABC’s “This Week.”

The U.S., he said, was “paying a terrible price” with the surge in Covid-19 cases because about half the country hasn’t been fully vaccinated, and about 90 million eligible people haven’t gotten a single dose.

The highly contagious Delta variant is prompting some people to reconsider past reluctance or resistance to vaccine mandates, including Randi Weingarten, president of the American Federation of Teachers, which is the nation’s second-largest teachers union.

“As a matter of personal conscience, I think that we need to be working with our employers, not opposing them on vaccine mandates,” Ms. Weingarten said on NBC.

Her union previously opposed unilateral vaccine requirements for teachers, but she said Sunday the organization would reconsider that stance this week, as students across the country prepare to return to the classroom, many of them too young to be inoculated.

“I do think that the circumstances have changed and that vaccination is a community responsibility, and it weighs really heavily on me that children under 12 can’t get vaccinated,” she said.

“Our number one goal is to get schools open,” Ms. Weingarten later wrote on Twitter. “Vaccines and COVID mitigation policies are key to doing that. We’re going to keep working to make sure schools are safe and welcoming in the fall.’’

Legal challenges to mandates are moving through the courts. Last week, a group of students at Indiana University filed an emergency appeal with the Supreme Court that seeks to block enforcement of the school’s vaccine requirement ahead of the coming semester. Lower courts rejected the request, noting among other things that the school policy does allow for exemptions, coupled with masking and testing.

A number of local entities and employers have come out in recent days with vaccine mandates for workers, including in nursing homes, a frequent source of Covid-19 cases.

United Airlines on Friday said it would require its 67,000 U.S. employees to be vaccinated this fall, the first major airline to take such a step. Other companies that recently announced full or partial vaccine mandates include Tyson Foods Inc., Microsoft Corp. and Walmart Inc.

Write to Brent Kendall at brent.kendall@wsj.com