WASHINGTON — The Biden administration and Michigan’s Democratic governor are locked in an increasingly tense standoff over the state’s worst-in-the-nation coronavirus outbreak, with a top federal health official on Monday urging the governor to lock down her state.

As the governor, Gretchen Whitmer, publicly called again for a surge of vaccine supply, the director of the Centers for Disease Control and Prevention said at a White House news conference that securing extra doses was not the most immediate or practical solution to the outbreak. She said that Michigan — whose metro areas include 16 of the 17 worst outbreaks in the nation — needed to enact shutdown measures to stamp out the crush of infections.

“The answer is not necessarily to give vaccine,” said the director, Dr. Rochelle Walensky. “The answer to that is to really close things down, to go back to our basics, to go back to where we were last spring, last summer, and to shut things down.”

Michigan’s outbreak — driven by a highly infectious virus variant, loosened restrictions, travel, youth sporting events and uneven compliance with the remaining rules — is by far the worst in the country. The state is averaging seven times as many cases each day as it was in late February, and hospitalizations have roughly doubled in the past two weeks. Nonetheless, Ms. Whitmer has stopped well short of the far-reaching shutdowns that made her a political lightning rod last summer, with armed protesters storming the Statehouse to demand an end to coronavirus restrictions.

The Biden administration, however, has held fast to distributing vaccines by state population, not by triage, shying away from anything that could look like inequitable distribution or political favoritism at a time when vaccine supply remains tight in many places.

“It’s important to understand how we’ve approached vaccine distribution from the beginning: It’s done with equity in mind. It’s done with the adult population in mind,” the White House press secretary, Jen Psaki, said Monday. “We don’t pick by our friends. We don’t pick through a political prism.”

Michigan’s renewed fight with the virus was a warning for other states seeing new increases in cases and could have far-reaching consequences. Reports of new cases have increased by 45 percent in Illinois over the past two weeks, with especially high infection rates around Peoria. And as new, more contagious variants spread, caseloads are rising in Minnesota, Pennsylvania and several other states.

In an interview on Sunday with CBS’s “60 Minutes,” the Federal Reserve chair, Jerome H. Powell, said the American economy had “brightened substantially” as more people have been vaccinated and businesses have reopened. But he cautioned that “there really are risks out there,” specifically coronavirus flare-ups, if Americans return to normal life too quickly.

“The principal risk to our economy right now really is that the disease would spread again more quickly,” Mr. Powell said.

In recent days, Ms. Whitmer, an ally of President Biden’s, has diverged repeatedly from the president, asking him in a private call last week for extra vaccines, and, after being turned down, continuing to press her case in public that vaccination is the answer.

Bobby Leddy, a spokesman for Ms. Whitmer, said the state was suffering not from a failure of policymaking, but from the new variants that are more contagious and from Michiganders who are not complying with the governor’s orders. “Which is why it’s important for us to ramp up vaccinations as quickly as possible,” he said.

Ms. Whitmer was joined in the call for more vaccines by Representatives Fred Upton, a Republican, and Debbie Dingell, a Democrat, who sent a letter last week to Mr. Biden pleading for extra doses for their state.

Ms. Whitmer, who called last week for voluntary pauses to indoor dining, youth sports and in-person high school, said on Monday that she planned to extend existing restrictions on in-person officework for six more months. She has also appealed to Michigan residents to take more “personal responsibility,” language that echoed Republican governors and contrasted sharply with her own response to earlier surges.

White House officials have said they are working with Michigan to help the state use the doses it still has on shelves. Eighty percent of those delivered so far have been administered, according to data reported by the C.D.C.

Andy Slavitt, a White House pandemic adviser, said on Monday that instead of playing whack-a-mole by rushing vaccines to hot spots, the federal government was working to help Michigan more efficiently administer the doses it has.

“We know there are appointments available in various parts of the state,” he said, “and so that means that we have excess vaccine in some parts of the state.”

Mr. Slavitt said that the federal government had also offered to send Michigan extra supplies of monoclonal antibody treatments for people with Covid-19, and that it was sending more tests to the state and was helping to set up more test sites. He added that a team from the C.D.C. was working in Michigan, as were 140 new vaccinators from the Federal Emergency Management Agency.

Ms. Whitmer, who spoke on Monday after touring a vaccination site in Ypsilanti, said she was grateful for the offer of additional therapeutics and personnel. But, she said, “we have been working incredibly hard to make sure we get what we need to get people vaccinated, and that includes lobbying for additional vaccines to come into Michigan.”

Federal health officials made clear they would not be grabbing doses from some states to rush to hot spots like Michigan. That would not be feasible, in part because the administration would not be able to guarantee how well the retrieved doses were stored once delivered.

“We’re not in a place, nor will we be, where we take supply from one state to give it to another,” Ms. Psaki said at her briefing on Monday at the White House.

Even though some doctors and public health officials in Michigan have called for new lockdowns, there is a bipartisan sense that another lockdown would be a political nonstarter, especially with the legislative Republicans who hold majorities at the Capitol in Lansing.

“She knows the possible better than I do,” said Mayor Christopher Taylor of Ann Arbor, a Democrat who said he had confidence in the governor and backed her call for extra vaccines. “She is conducting, all the time, a balancing test of listening to the science and engaging what we’re able to actually get through Lansing.”

State Representative Steve Johnson, a Republican, said he doubted that many people would comply with a lockdown order. “For her to try to continue those measures would have been political suicide,” he said.

During previous surges in Michigan, Ms. Whitmer shut down businesses and schools as she saw fit, drawing intense protest from Republicans in the state, who viewed her as an avatar of government overreach. The state still has a mask mandate in place and strict capacity limits on a number of activities.

The clash between the Biden administration and Michigan also revealed disagreement among public health experts, some of whom have said Michigan’s outbreak requires a fast and drastic shift in how doses are distributed.

“What you see is that Michigan is an outlier that’s profound,” said Dr. Eric Topol, a professor of molecular medicine at Scripps Research in La Jolla, Calif. “This is a precedent in the country. It’s about plasticity, flexibility in responding, in being able to pivot.” He added that tens of millions of doses were sitting unused across the country, and “in some states, you can’t even give them away, like Mississippi and Alabama.”

Vaccines could have been surged to Michigan weeks ago when signs of its new wave of infections were appearing, he said, like signs that are now showing up in other states, such as Minnesota.

“We have this incredibly powerful tool, and we’re not using it,” he said. “And it’s just an outright shame.”

But other public health experts have said that because it takes weeks for full protection from vaccines to kick in, the effects of sending extra doses to the state would take too long to see.

“I think if we tried to vaccinate our way out of what is happening in Michigan, we would be disappointed that it took so long for the vaccine to work, to actually have the impact,” Dr. Walensky said, adding, “If we vaccinate today, we will have, you know, impact in six weeks, and we don’t know where the next place is going to be that is going to surge.”

Dr. Tom Frieden, a former C.D.C. director, said wearing masks indoors and avoiding indoor events were more immediate remedies.

Someone is not considered fully vaccinated until two weeks after the second dose of the vaccines made by Moderna or Pfizer-BioNTech, or after the single-dose shot made by Johnson & Johnson.

“If you’re trying to vaccinate now to address the transmission that’s occurring right now, that’s just not going to work. The timeline doesn’t match up,” said Dr. Céline Gounder, an infectious disease expert and a member of Mr. Biden’s transition’s Covid-19 task force.

Dr. Mahshid Abir, an emergency physician at the University of Michigan and a senior policy researcher at the RAND Corporation, said that with hospitals overwhelmed, she was sympathetic to both Ms. Whitmer’s predicament and the White House’s position on sending more vaccines because the surge in the state was already out of control. “The ship has sailed on that,” she said.

But another lockdown remains unlikely. With 23 percent of Michigan residents fully vaccinated, and 35 percent of residents with at least one shot, Ms. Whitmer said it was time for a different approach.

“I believe government’s role is, when we can’t take action to protect ourselves, the government must step in,” she said Monday. “We’re in a different moment.”

Noah Weiland reported from Washington, and Mitch Smith from Chicago.