He also defended calling the crisis a "hoax" at a rally hours before a woman in Washington state lost her life from the virus, which has killed more than 2,000 people across the world.
Asked at a news conference on government plans to protect Americans about whether it was a mistake to use such language, Mr Trump said his comment was aimed at the Democrats' claims about his response.
He said: "No no no. 'Hoax' referring to the action they take to try and pin this on somebody, because we've done such a good job."
Mike Pence, who is heading the taskforce on coronavirus, announced a series of new travel restrictions which the administration hopes will help limit the spread of the deadly virus.
The existing travel ban on Iran will be extended to include foreign nationals who have travelled there in the previous 14 days, while US government advice against travel to areas of Italy and South Korea will be strengthened.
Mr Trump also said he was considering travel restrictions on the southern border with Mexico, although there have been no known cases of coronavirus transmission there.
The president insisted that the markets would "come back", following historic losses over the past week over fears that the crisis could hit the economy, both through supply lines but also through a drop in consumer spending as people worried about catching it stayed at home.
A sombre President Trump, who had been widely criticised for not appearing to take the crisis seriously at his previous news conference, allowed public health expert Dr Anthony Fauci and his health secretary Alex Azar to speak, and told reporters: "We'll be doing this quite often and keep you abreast of all the facts."
Speaking of the person who died in Washington, he said that she was a "wonderful woman" in her late 50s who was at high medical risk, although officials in Washington later said the patient who died was a man. An official in the West Coast state said there were a number of other people infected.
The president encouraged Americans not to alter their daily routines, saying the country was "super prepared" for a wider outbreak, adding: "There's no reason to panic at all.
"You're talking about 22 people right now in this whole very vast country. I think we'll be in very good shape."
Dr Fauci denied reports that he had been prevented from speaking directly to the media, following the announcement that all experts would have to go through Mr Pence. He said this was a "misrepresentation".
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