Lloyd J. Austin III, the defense secretary, has written a letter to seven Republican governors, rejecting their requests for exemptions from coronavirus vaccination mandates for their states’ National Guard troops.

The rejection — sent to the governors of Alaska, Oklahoma, Texas, Idaho, Mississippi, Nebraska and Wyoming, who have all sought to allow their guard troops to refuse the vaccine without consequences — sets the stage for a potential legal battle.

“In making the decision to require vaccination against Covid-19 for service members,” Mr. Austin wrote: “I considered the thousands of hospitalizations and the hundreds of deaths among service members, civilians, and their families related to Covid-19. Covid-19 takes our service members out of the fight, temporarily or permanently, and jeopardizes our ability to meet mission requirements.”

“Failure by a member to do so will lead to a prohibition on participation in drills, training, and other duty conducted under Title 32 and will jeopardize the member’s status in the National Guard,” Mr. Austin added.

Federal officials have long said that governors have no legal standing to allow Guard members to refuse to comply with the military’s vaccine mandate. State officials and some legal experts, however, believe that unless National Guard members are federally deployed, they are under the jurisdiction of the governor of their state and therefore not subject to federal mandates. It will be up to the governors to see how far they wish to push this dispute.

Roughly 97 percent of the 1.3 million active-duty service members in the United States military have had at least one dose of a coronavirus vaccine, but the proportion is much lower for the Guard and Reserve, who still have time to comply before their deadline later this year. The deadlines for active-duty forces passed late last year, with fewer than 100 members of the military being separated for failure to comply. Not a single soldier in the Army, for instance, has left so far over refusal to get a shot.

Exemptions from the mandate have been few in number, and often temporary. Two members of the U.S. Marine Corps were recently given religious exemptions from the Pentagon’s vaccine mandate, the first of their kind since the mandate was announced last summer.