A U.S. senator from Washington, the state hardest hit by the spreading coronavirus pandemic, on Thursday requested an investigation into the federal government’s failure to deliver badly needed tests that detect the new virus, according to the lawmaker.

“I am still hearing from people that they go to their health-care provider and they say, `Gosh, we don’t have any test kits available,’ ” said Democratic Sen. Patty Murray in an interview Thursday.

In a letter to the Department of Health and Human Services Inspector General, Ms. Murray asked the watchdog to investigate the federal health department’s efforts to “develop, deploy, and analyze diagnostic tests for the 2019 Novel Coronavirus.”

Tesia Williams, a spokeswoman for the HHS inspector general said the agency is reviewing the letter. “We continue to monitor the situation around the Covid-19 outbreak and have plans to conduct a number of reviews regarding HHS’s response and planning efforts,” she said.

The Wall Street Journal reported Wednesday that federal health agencies’ testing efforts were hampered by a series of missteps. A Food and Drug Administration requirement deterred private testing, lab operators said. The Centers for Disease Control and Prevention bungled a test developed in a government lab, according to documents and interviews. Top leaders failed to ensure adequate supplies and have struggled to turn around the flawed rollout despite weeks of public pressure, the Journal found.

An FDA spokeswoman earlier said a 2004 law required the agency to take the action labs have criticized, and the CDC earlier said it is still investigating the cause of the glitch. HHS said it had engaged with the private sector early on.

On Thursday, President Trump took issue with the Journal’s reporting during a televised briefing from the White House. “We’ve done a phenomenal job on this,” he said of the testing program.

A White House spokesman didn’t immediately respond to requests for comment.

Washington state had the first confirmed new coronavirus case. A man went to an urgent care clinic on Jan. 19 with cough and fever after traveling to Wuhan, China.

The virus has since hammered the Seattle area, after it was identified at the Life Care Center of Kirkland, a nursing home, in late February. The local public health department said Wednesday that of the county’s 56 deaths linked to coronavirus as of that time, 35 were associated with Life Care Center of Kirkland.

The nursing home operator didn’t immediately respond to a request for comment.

As of Thursday, Washington state has 1,187 confirmed cases of Covid-19, the disease caused by coronavirus, about 10% of the national total, according to data compiled by Johns Hopkins University.

Ms. Murray said in the interview that she raised alarms with the administration about the testing shortages since mid-February, as more cases began appearing in her state. “All of my conversations with them was, `Not to worry, by Friday we’ll have ‘X’ amount,’ ” she said. “The numbers just seem so made up.”

She said she is hearing from constituents that supply shortages are impeding both their ability to do tests and provide health care in general, ranging from masks to reagents, which are chemicals used in the testing process. “People are just literally in tears telling me they don’t have it available, even at the epicenter of where it started,” she said.

The letter seeking the investigation says, “It is clear HHS’s grave errors in managing every aspect of the testing process—from development to deployment to analysis to communication—have undermined the country’s ability to mitigate the spread of Covid-19.”

Ms. Murray specifically asked the HHS inspector general to answer open questions about why the CDC’s initial test didn’t work as expected, how HHS assessed its own testing capacity and the private sector’s, and what impact delays in ramping up a robust testing program has had on the country.

“I want answers,” Ms. Murray said. “I want to make sure when and if we get through this we fix the problem.”

Write to Christopher Weaver at christopher.weaver@wsj.com

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