Americans "should be prepared that they're going to have to hunker down significantly more than we as a country are doing" to fight the growing COVID-19 outbreak, Dr. Anthony Fauci, director of the National Institute of Allergy and Infectious Diseases, said Sunday on NBC News' "Meet the Press."

Asked whether the U.S. should consider a 14-day national shutdown similar to those in Europe, he said: "I would prefer as much as we possibly could. I think we should really be overly aggressive and get criticized for overreacting."

The U.S. has surpassed 2,000 confirmed cases of the coronavirus, although the actual number of those infected could be much larger because testing isn't yet widespread. At least 60 people have died in the U.S. because of complications from the virus.

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Fauci said the goal now is to "blunt" the curve of confirmed cases, keeping the number of those infected low enough so as to not overwhelm the U.S. hospital system.

"If you let the curve get up there, then the entire society is going to be hit," he said.

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In response to the crisis, some states and municipalities have already banned large gatherings and closed schools, sports leagues have suspended play and a handful of elections and state party conventions have been postponed. The presidential campaign is going virtual, with President Donald Trump, Sen. Bernie Sanders, I-Vt., and former Vice President Joe Biden having canceled rallies and other in-person events.

Fauci said he has let the Trump administration know his feelings on that, adding that "they listen and," for the most part, "go with what we say."

Of the elderly and those with underlying conditions, Fauci said, "They should really hunker down."

An NBC News/Wall Street Journal poll released Sunday found that 60 percent of Americans believe the worst is yet to come, with 40 percent saying their day-to-day lives will change.

Of Trump's handling of the outbreak, 45 percent of voters said they approved.

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On the initial lack of testing for the virus, which has led to widespread criticism of the administration's handling of the outbreak, Fauci said there has been a "sea change," adding that "early on, we weren't in a situation where we could get the tests out in a broad way." He said that would soon be changing.

Last week, Fauci said at a House hearing that the system was "not really geared to what we need right now" regarding the test kits, saying: "That is a failing. Let's admit it."

Asked Sunday about cases of people having contracted severe symptoms overseas even though they are outside the stated risk groups, Fauci said researchers will want to "make sure the virus hasn't changed," adding that while younger people are much less at risk, "that doesn't mean they're not going to get infected, and then they'll be infecting older people."

Fauci called for elective surgery to be delayed, reserving hospitals for those who absolutely need to be there.

If you feel possible symptoms of the virus, Fauci said, you should contact a physician and get instructions on how to be tested, "but stay home."

Speaking on CNN's "State of the Union," Fauci said there's a chance that thousands, if not millions, of Americans could die because of the outbreak.

"It's possible because, when you do a model, you have a worst-case scenario, the best-case scenario, and the reality is, how you react to that will depend where you're going to be on that curve," he said.