(CNN) At a time when there are not enough coronavirus tests to meet demand in hospitals across the United States, those who can afford it and have the right connections can still get tested.

In the Los Angeles area, at least two so-called "concierge doctors" tested hundreds of people, or provided them tests to take home, for between $250 and $600 per test.

The Medical Board of California is "aware of some doctors doing this" and is looking into the allegations, spokeswoman Veronica Harms told CNN.

"If we are made aware of any of our licensees engaging in such activities, whether it be a doctor, nurse, physician assistant, etc., we will investigate the situation and take appropriate action," Harms said in a statement.

The US Food and Drug Administration did not provide immediate answers to CNN's questions about the legality of selling tests or letting patients take tests home for later use.

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Dr. Arthur Caplan, founder of the New York University School of Medicine's division of medical ethics, said no laws or regulations were necessarily violated, but the practice lays bare the vast inequality in the American health care system.

Describing it as "one of the grossest scandals of America that class drives access to resources in an epidemic," he said state and federal resources haven't properly been put into action.

"It's a system failure. It would also be nice if somebody in Washington (DC) had said, 'Here's how we're going to fairly distribute tests, ventilators, protective gear,' instead of left it up to the market," he said.

Doctor did nothing wrong, he says

Long before the novel coronavirus pandemic, Caplan said, there were people who were poor, didn't have insurance or had enormous co-pays for their insurance, "but now, people with insurance, people who are middle-class, they find themselves wondering, 'I wonder if the rich are gonna push me aside?' And the answer is: Yes, they can."

Where government is supposed to be accountable to the people, "the free market, the invisible hand, doesn't care who it kills, has no ethics," he said.

Dr. Jay Gordon, a Santa Monica pediatrician, insists he did nothing wrong by giving his patients tests to take home. In fact, he said, he believes he increased demand -- and thus production -- by selling tests to well-heeled patients.

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"It's not that I took tests away from people because the tests weren't available to them," the doctor told CNN.

He doesn't know how many tests he sold -- at least 300 but maybe as many as 500 -- for $250 a pop, he said. The first 100 sold in 10 minutes, Gordon said. He's donating the profits to charity, he said.

"Use them if you have symptoms or if you have a problematic exposure," Gordon said he told his patients.

He further advised them to hold on to them and not to use them right away, he said.

$50 test, $600 bill

Dr. David Nazarian is a primary care physician who has a "concierge" practice in Beverly Hills offering "boutique" or "executive" health care to his patients.

"We try to provide the best care that we can. It's more of a one-on-one experience where we're there for patients when they need us," Nazarian said.

In February, when the coronavirus outbreak began making headlines, Nazarian contacted test manufacturers LabCorp and Quest Diagnostics to order Covid-19 test kits for his patients as part of his practice's role in anticipating "certain factors that may impact health care," he said.

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"It's no different than how I approach medical advances and leading technology on the market, so that's why we had test kits where some other places maybe had difficulty finding them."

He administers the tests only to patients who fit federal health officials' criteria for testing, he said.

Patients are billed $50 for the actual test, which he performs only after an initial consultation, which costs between $300 and $400. He then creates new charts for the patients and schedules follow-up visits once the test results arrive.

The total bill, once the results are in, is about $600, he said.

States take action amid shortages

This all comes as governors across the country are pleading for more test kits for potential Covid-19 patients.

"We have a desperate need for the testing kits," said Gov. Jay Inslee of Washington , the site of the first deadly hotspot once the virus arrived in the United States.

In addition, one of the nation's largest commercial labs has faced a backlog of tests that has delayed results in some cases for up to 10 days.

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Meanwhile, health officials on both coasts have asked physicians to prioritize who they test amid the shortage.

In March, the Los Angeles Department of Public Health advised doctors not to test those experiencing only mild respiratory symptoms unless "a diagnostic result will change clinical management or inform public health response."

Soon after, the New York City Department of Health directed all health care facilities to immediately stop testing nonhospitalized patients for Covid-19.

'I was creating a nice demand'

The ethics of selling test kits aside, the FDA "has not authorized any test that is available to purchase for testing yourself at home for COVID-19," it says.

The guidance appears more aimed at companies claiming they have manufactured tests approved for at-home use, rather than doctors selling hospital test kits to take home.

The FDA hopes to make home testing available at some point, it says.

Gordon ordered his tests, he said, from Curative , a Bay Area startup founded earlier this year as a manufacturer of sepsis tests. It pivoted to Covid-19 tests after the outbreak reached the US and began making tests for drive-through sites organized by Los Angeles authorities.

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"If anything, I was creating a nice demand so that there would be more production," Gordon said.

More than 10,000 of the company's tests have been administered, spokeswoman Marissa Shapiro said.

"Once the company realized where the demand was going to be coming from, which obviously was cities and first responders, their attention was directed toward those entities," Shapiro said.

Curative does not distribute at-home testing, Shapiro told CNN, but the company's website promotes the idea of "self-administered testing" to stem the spread of the virus and notes that its Covid-19 test "can be self-administered at home" with results as quickly as 24 hours.

Doctor says he offered all patients tests

Gordon, who appeared last year on HBO's "Real Time with Bill Maher" deriding toxic metals in baby foods and vaccine laws, told CNN his patients are "more likely to have discretionary funds."

Some of his patients pay a monthly fee, he said, to have guaranteed 24-7 access to his services "within three to 10 seconds."

The tests he is selling are too expensive for some people, he acknowledged, but he offered them to all of his patients, including those of his four practice partners.

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"The tests were not offered to just wealthy people," he said.

The tests cost him $200 each, and the profit from the sales was donated to a Christian medical mission that operates in Africa, he said.

Until this week, the Los Angeles County Department of Public Health prioritized testing for those with symptoms, health care workers, residents of long-term care facilities, paramedics and others in high-risk situations, but Mayor Eric Garcetti announced Monday there are "no longer any limits" on who can apply to be tested.