Anthony Fauci, director of the National Institute of Allergy and Infectious Diseases, speaks during a Coronavirus Task Force news conference at the White House in Washington, D.C., U.S., on Friday, April 10, 2020.

White House health advisor Dr. Anthony Fauci on Sunday expressed "cautious optimism" that the deadly coronavirus outbreak is slowing down in the United States, and said that parts of the country may start to reopen as soon as May.

Fauci, the director of the National Institute of Allergy and Infectious Diseases, warned on CNN's "State of the Union" that the entire country won't suddenly turn back on like a "light switch."

But there are "indications" that some of the metrics used to gauge the crisis "are starting to level off" in some areas, he said.

Asked when parts of the U.S. could start to relax some of their strict social-distancing measures, Fauci said that the process could probably begin "at least in some ways, maybe next month."

Fauci's hopeful remarks came less than a day after the U.S. counted more than 20,000 deaths from the disease, surpassing Italy as the nation with the highest number of recorded deaths in the world. More than 530,000 people in the U.S. have tested positive for COVID-19, according to data from Johns Hopkins University.