During his confirmation hearings last week, Lloyd J. Austin III, the secretary of defense, said he would increase military support to manage the pandemic. On Thursday, Max Rose, Mr. Austin’s senior adviser for Covid-19, said his first topic of conversation in meetings with senior leaders had been making this the “No. 1 priority.”

Sending troops to help set up sites, assist with logistics and even put shots in arms is something the Defense Department is “actively considering,” Mr. Rose said. He declined to provide specifics, saying that Pentagon officials would be reviewing the request from FEMA carefully.

“We are obviously going to source this request,” said John F. Kirby, the Pentagon press secretary, noting that, after a review, it would most likely be filled with more National Guard, reserve and active-duty troops.