The White House, meanwhile, is still debating which types of tests should be sent to which regions and how much to focus on testing Americans to see who may have developed immunity to the disease.

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“Unfortunately, states really are on their own,” said Partners in Health medical director Joia Mukherjee, whose group is working with Massachusetts to develop the country’s most extensive contact tracing network to track infected patients’ interactions with others. “It’s problematic at best and egregious at worst, because some states have more resources than others; some states have more leadership than others.”

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Public-health experts say that widespread testing for the novel coronavirus is key because it would determine who is infected and needs to be isolated, as opposed to ordering the entire population to stay at home. The development and use of antibody tests, meanwhile, could identify those with immunity to the virus who are able to return to work, school and everyday activities.

Coronavirus testing in the United States started late and has been beset by a string of problems, including bureaucratic delays and ongoing shortages of materials. While testing has accelerated now that private companies are processing samples, even the most aggressive states have tested just a small fraction of their residents. New York has tested 1,645 of every 100,000 people and Utah 1,043 of every 100,000, but Texas has tested just 297 and Georgia 381. On a per capita basis, the United States lags behind several other countries in the rate of testing.

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“Testing is getting more available,” said Scott Becker, executive director of the Association of Public Health Laboratories. “But the demand is absolutely still outstripping supply.”

In recent days, the White House coronavirus task force has begun debating what a national testing strategy would look like, according to several senior administration officials. Leading that effort are Deborah Birx, White House coronavirus response coordinator, and Brett P. Giroir, the assistant secretary for health at the Department of Health and Human Services.

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At a Monday task force meeting, according to a participant who spoke on the condition of anonymity to discuss internal deliberations, Birx and Giroir debated where to send the newest coronavirus tests — a version produced by Abbott Laboratories that can deliver results on-site in as little as five minutes, as opposed to tests that can take hours and must be processed by a laboratory.

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One senior administration official said the government hopes to devise recommendations for different populations in rural, suburban and urban areas. Officials also plan to pick different spots in the country to collect data so they can be better prepared to respond in the fall, when the virus could make a return.

The administration has launched a separate effort to test for antibodies, using a process known as serology testing, to identify people with probable immunity to the virus. While these tests could be critical for longer-term decisions about lifting stay-at-home orders, there is a debate about the accuracy of the tests and how they should be used.

At a White House briefing Monday evening, Giroir said an interagency working group was closely examining the new serology tests because “we have reason to believe that not all of them are going to perform well.”

In late March, the FDA adopted an “enforcement discretion” policy on serology tests, saying they could be sold without prior authorization as long as the companies validated their data, notified the agency and made clear the tests should not be used as the “sole basis” for diagnosing active infections and had not been approved by the agency.

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Since then, several tests of dubious value have appeared on the market, said the lab association’s Becker, who also said he recently shared his concerns about the “wild, wild West” environment with Giroir and FDA Commissioner Stephen Hahn.

Giroir told reporters he was confident this effort would give the United States a reliable tool by next month to determine who had been exposed to the virus and could reenter society safely.

“We will have millions on the market by May, in a sophisticated way and a prospective way, that we get the surveillance we need,” he said. “We can test people to see if they have been exposed, [are] immune and [can[ go back to work. And a combination of that kind of test, with the current kind of test we have now, is how America opens back up.”

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Officials have also indicated they want major manufacturers, such as Roche Diagnostics, to provide serology tests.

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In the meantime, Joe Bresee, deputy incident manager for CDC’s pandemic response, said the agency has begun testing blood samples from people in coronavirus “hot spots” to determine whether those people, whether they know it or not, may have antibodies that give them some immunity.

“We hope it will give a sense of the area of water under the iceberg,” Bresee said.

But some experts warn that the promise of these serological tests, which use blood samples and are relatively simple to administer and process, may be overstated. And they emphasize that the federal government must invest resources now to create the capacity for widespread diagnostic tests so the nation is prepared in the coming months to try to head off a resurgence of the virus later in the year.

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“Coronavirus testing should be as ubiquitous as strep testing, and maybe more so,” said Scott Gottlieb, who headed the Food and Drug Administration early in Trump’s presidency and is informally advising the White House as well as the governors of Massachusetts and Maryland.

Gottlieb, in a new paper written with experts including Mark McClellan, FDA chief during the George W. Bush administration, argues that the nation needs to create a massive testing and surveillance infrastructure by late May to try to get ahead of the epidemic and prevent an autumn comeback. He said the government should focus on the mass distribution of rapid point-of-care tests to hospitals, clinics and doctors’ offices.

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Some Trump officials said it is too late to create a national testing strategy and that it should have been done weeks, if not months, ago. “We were a month behind on getting testing set up, and now we’re just trying to ramp up testing every single day,” said one official, speaking on the condition of anonymity to be frank.

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In the meantime, several states have adopted their own plans.

In Utah, the University of Utah and Intermountain Healthcare worked with ARUP Laboratories to develop mobile testing sites. On Thursday TestUtah.com was launched to allow residents to fill out an online survey to determine whether they need coronavirus tests and to book appointment to be tested. The venture is similar to one Trump described a month ago that has not materialized nationwide.

California Gov. Gavin Newsom (D) assembled a task force Saturday, co-chaired by assistant director of the California Department of Public Health, Charity Dean, and by Blue Shield of California President and CEO Paul Markovich.

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One member of the state task force, who spoke on the condition of anonymity to avoid disrupting help from the federal government, said California pressed ahead with initiatives such as establishing eight testing hubs in collaboration with public and private universities and Kaiser Permanente because it had no choice.

“We’re making up our strategy because the federal government has been completely out to lunch,” the task force member said. “We can’t mobilize a health response without knowing how this disease is spreading.”

Connecticut’s governor has also formed a task force to develop a testing strategy. One focus, called Project Haystack, is aimed at detecting contagious individuals.

“The tragedy that we find ourselves in is that we didn’t do the testing early enough,” said task force member David Scheer, president of a venture capital and advisory firm.

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And in Maryland, where Gov. Larry Hogan (R) initially requested a testing site, state officials are now working with Gottlieb to implement their own plan. “The request for the federal testing site is no longer our top priority,” Hogan said in a recent interview.

Correction: A previous version of this article referred to Roche Diagnostics as a laboratory. It is a manufacturer.