The nation’s largest teachers’ union on Thursday offered its support to policies that would require all teachers to get vaccinated against Covid or submit to regular testing.

It is the latest in a rapid series of shifts that could make widespread vaccine requirements for teachers more likely as the highly contagious Delta variant spreads in the United States.

“It is clear that the vaccination of those eligible is one of the most effective ways to keep schools safe,” Becky Pringle, president of the National Education Association, said in a statement.

The announcement comes after Randi Weingarten, the powerful leader of the American Federation of Teachers, another major education union, signaled her strongest support yet for vaccine mandates on Sunday.

Ms. Pringle left open the possibility that teachers who are not vaccinated could receive regular testing instead, and added that local “employee input, including collective bargaining where applicable, is critical.”

Her union’s support for certain requirements is notable because it represents about three million members across the country, including in many rural and suburban districts where adults are less likely to be vaccinated. Overall, the union said, nearly 90 percent of its members report being fully vaccinated.

Still, any decision to require vaccination for teachers is likely to come at the local or state level. And even with their growing support, teachers’ unions have maintained that their local chapters should negotiate details.

“We believe that such vaccine requirements and accommodations are an appropriate, responsible, and necessary step,” Ms. Pringle said on Thursday. She added that “educators must have a voice in how vaccine requirements are implemented.”

California has ordered all teachers and staff members to provide proof of vaccination or face weekly testing, an order that applies to both public and private schools. Hawaii is requiring all state and county employees to be vaccinated or be tested, including public-school teachers. And Denver has said that city employees, including public school teachers, must be fully vaccinated by Sept. 30.