Calling in to the NY1 cable channel later in the day, Mr. Cuomo reiterated that position.

“There’s not going to be any quarantine, where we contain people within an area, or we block people from an area,” he said. “Individual mobility is what we’re all about. There’s not going to be any you-have-to-stay-in-your-house rule.”

Mr. de Blasio also said on Tuesday that the city would be able to test 5,000 people a day for coronavirus, a major increase from the current capacity of several hundred people a day. And he said that alternate-side parking rules would be suspended for at least a week.

As of Tuesday, more than 1,500 people in New York State had tested positive for the virus, up from 950 the day before; 814 of the cases were in the city, including that of a New York Police Department officer assigned to the First Precinct in Lower Manhattan. (Seventeen officers who may have had contact with that officer were in self-quarantine.)

Fifteen people have died from the virus in New York State, including at least 10 in New York City.

“We are certainly going to have thousands of cases next week,” Mr. de Blasio said. “It is not that long before we hit 10,000 cases, that is a true statement.”