WASHINGTON — Vice President Mike Pence said the Centers for Disease Control and Prevention has lifted all restrictions on testing for the coronavirus and would be releasing new guidelines to fast-track testing for people who fear they have the virus, even if they are displaying mild symptoms.

“Today we will issue new guidance from the CDC that will make it clear that any American can be tested, no restrictions, subject to doctor’s orders,” Pence told reporters at the White House. “Any American can be tested. We’re removing that barrier.”

The Trump administration has struggled to project confidence and progress in the fight against the coronavirus and simultaneously prepare Americans for what could be a difficult struggle. The slow start in testing for the virus has been particularly vexing.

Officials at the White House emphasized that the supply of tests may not fully meet demand until the coming weeks, but they said around 2,500 test kits for more than 1 million individual tests could be available by the end of the week, and that an additional 1 million tests could be manufactured weekly after that.

“We are trying to meet that demand with increased capacity,” Dr. Stephen Hahn, the commissioner of the Food and Drug Administration, told reporters during the briefing. “Right now, I’d say we’d need more capacity.”

On Saturday, the FDA gave laboratories and hospitals around the country the go-ahead to conduct tests that had been limited to those analyzed by the CDC, which had been enforcing strict guidelines about who was allowed to be tested. Pence’s announcement Tuesday loosened those regulations further.

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Hahn said at a Senate hearing that the CDC was working with a private manufacturer to drastically increase the testing capacity of laboratories across the nation.

“Our expectation in talking to the company that is scaling this up is that we should have the capacity by the end of the week to have kits available to the laboratories to perform about a million tests,” he said.

When Hahn was asked to clarify, he said he was hearing from private manufacturers that 2,500 test kits could be available by the end of the week, with each kit capable of 500 tests. Later in the day, Pence repeated that number to reporters, saying the kits amounted to 1.5 million tests.

“This is a dynamic process,” Hahn said. “Every day we’re hearing from additional manufacturers.”

But under questioning from senators in both parties, Pence, who is leading the administration’s coronavirus response, and other top health officials were unable to satisfy lawmaker concerns.

“They could not answer how soon people would be able to get the tests,” said Sen. Chuck Schumer of New York, the Democratic leader, emerging from a lunchtime briefing with Pence and top health officials. “We need an on-site test, not a test that has to be sent far away and sent back, and we need to know where people can get it, how people can get it.”

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Sen. Bill Cassidy, R-La. and a medical doctor, agreed, “there was concern about the testing — a lot of concern about testing.”

“The point was made in there that everybody who gets sick is above age 50 almost,” Cassidy continued, “and nobody has been dying less than age 20, but that doesn’t mean they’re not infected.”

Democrats also raised concerns with the officials about making sure the test is “not so costly that people don’t get it,” Schumer said. “And there were no answers to those vital questions.”

The issue of vaccine affordability is a final sticking point in negotiations on an emergency aid package to respond to the virus, which lawmakers and staff members said they hoped to unveil late Tuesday. Schumer said the package, which is expected to be at least $7 billion, would include funding to state and local agencies responding to the virus, as well as funding for equipment and the development of a vaccine.

The package, which has been quickly cobbled together over the past few days, is expected to be far larger than the White House’s initial proposal: $1.25 billion in new funds, paired with a transfer of existing funds from other health programs.

House Democrats have said they hope to hold a vote on the emergency bill by the end of the week. Democrats want to ensure that drugmakers charge the government reasonable prices for treatments and vaccines once they come available.

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“We believe that the vaccine should be very cost-effective, and not be out of the reach of average folks,” Schumer said. “That’s one of the arguments that’s still being disputed.”

New infections in Westchester County, New York; San Mateo County, California; Wake County, North Carolina, and Fulton County, Georgia, since Monday evening made clear that the coronavirus was spreading in the sprawl of America’s largest urban centers and was no longer tethered to international travelers. In the United States, there have been at least 120 cases of coronavirus confirmed by lab tests as of Tuesday evening, and worldwide infections neared 93,000.

The number of tests that will actually be administered in the coming days could be substantially lower than the administration’s projection of 1 million by next week. A spokesman for the Department of Health and Human Services said on Monday that public health labs currently can test 15,000 people, and could test up to 75,000 by the end of this week.

The Association of Public Health Laboratories, which represents state and local government laboratories around the country, has said that its labs would be able to conduct about 10,000 tests a day when all of its 100 members that can perform testing are running. Scott Becker, the executive director of the lab association, said on Monday that labs can run about 100 tests per day. As of Monday, he said fewer than half of those labs were able to do so.

“All that matters is how many people you can test a day,” said Scott Gottlieb, who was the FDA commissioner between 2017 and 2019.

As they learned on Tuesday that the death toll in their state had reached nine, the two Democratic senators from Washington pressed Pence and the health officials about both the administration’s response and its message to the public, particularly about the availability of testing.

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“Our message to them was, be clear that they aren’t readily available, and the universe of people who have access to them is not large enough,” Sen. Maria Cantwell said. “Let’s figure out a more aggressive plan.”

The state’s other senator, Patty Murray, said she had been hearing from constituents who were desperately calling “every number they can find” for information, and cannot get through.

“I think there’s a lot of concern about honest, accurate reflection of what is real when you have something like this happening,” Murray said.

She criticized the Trump administration’s efforts to play down the effects of the virus.

“What I really feel strongly about is, we do not need Mick Mulvaney telling people don’t watch TV,” she added, referring to remarks made by the acting White House chief of staff. “We do not need the president saying this is fear mongering.”

Dr. Anthony Fauci, the director of the National Institute of Allergy and Infectious Diseases, told Politico in an interview on Friday that Americans should expect the outbreak to worsen.

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“I don’t think that we are going to get out of this completely unscathed,” he said. “I think that this is going to be one of those things we look back on and say, ‘Boy, that was bad.’”

Fauci also said that he would be truthful in his public pronouncements, even as President Donald Trump sought to minimize the virus’s impact.

“You should never destroy your own credibility. And you don’t want to go to war with a president,” Fauci told Politico. “But you got to walk the fine balance of making sure you continue to tell the truth.”