"It makes absolutely no sense and I don't think any serious governmental personality or professional would support it," Cuomo said.
Trump's proposed quarantine would restrict travel to and from the three states, which are some of the hardest-hit by the outbreak, as it emerged that 209 people died in New York state in the last 24 hours.
New York state now has at least 52,318 confirmed cases, nearly half the national total of more than 113,000.
In New York City alone, there are 29,766 confirmed cases, and there have been at least 450 deaths.
"Some people would like to see New York quarantined because it's a hotspot — New York, New Jersey maybe one or two other places, certain parts of Connecticut quarantined. I'm thinking about that right now," he said Saturday.
"We might not have to do it but there's a possibility that sometime today we'll do a quarantine - short-term - two weeks for New York, probably New Jersey, certain parts of Connecticut."
Trump said the possible quarantine would be "short-term" but that it would be "enforceable".
The president dismissed the idea that he would need to deploy the National Guard to ensure residents comply with the quarantine rules.
"We're not going to need that," he said.
"We're looking at it and will be making a decision. A lot of the states that are infected - they've asked me if I'd look at it so we're going to look at it. Maybe for a short period of time," Trump said.
"It would be for a short time" for parts of New York, Connecticut and New Jersey, he said.
However, when asked if he would shut down the New York City subway he said: "No, we wouldn't do that."
Cuomo said he had spoken with Trump earlier Saturday and the two had not discussed a possible quarantine. The governors of New Jersey and Connecticut also said that they had not spoken with Trump about a potential quarantine.
It was not clear whether Trump would be able to block road, air and sea travel out of a region that serves as the economic engine of the eastern United States, accounting for 10 per cent of the population and 12 per cent of GDP.
Some states have already imposed limits. New Yorkers arriving in Texas, Florida and Rhode Island face orders to self-isolate if they intend to stay.
The Rhode Island National Guard started going door to door on Saturday in coastal areas to inform any New Yorkers who may have come to the state that they must self-quarantine for 14 days.
Rhode Island Governor Gina Raimondo also expanded the mandatory self-quarantine to anyone visiting the state.
Raimondo also ordered residents to stay at home, with exceptions for getting food, medicines or going to the doctor, and ordered non-essential retail businesses to close Monday until April 13 to help stop the spread of the coronavirus.
She also directed realtors and hotel operators to include new requirements that any out-of-state residents must quarantine for 14 days in their purchase agreements.
State police set up a checkpoint on I-95 in Hope Valley on Friday where drivers with New York licence plates must stop and provide contact information and were told to self-quarantine for two weeks.
If New Yorkers don't comply, they face fines and jail time, Raimondo said, adding that that's not the goal.
"I want to be crystal clear about this: If you're coming to Rhode Island from New York you are ordered into quarantine. The reason for that is because more than half of the cases of coronavirus in America are in New York," Raimondo said, adding that it's not meant to be discriminatory.
The governors of Pennsylvania and West Virginia have also asked visiting New Yorkers to voluntarily self-quarantine.
New Hampshire Governor Chris Sununu on Saturday asked all visitors to his state who don't come for work reasons to voluntarily self-quarantine.
Trump said any New York-area lockdown would only apply to people leaving the region. It would not cover truckers making deliveries or driving through the area, he said.
US law gives the president the authority to restrict travel between states, legal experts said. But he would not be able to enlist local police to set up checkpoints along state lines, and it would be difficult to determine who would be allowed to get through, said Louisiana State University law professor Edward Richards.
"The logistics of deciding who is an essential person or essential cargo could shut down the ability to transport essential personnel and supplies," he said.
Even if it were possible, a New York-area lockdown might come too late for the rest of the country.
The number of coronavirus patients in California hospitals increased by more than one-third overnight, Governor Gavin Newsom said.
Officials in Louisiana, where Mardi Gras celebrations late last month in New Orleans fuelled an outbreak, reported 17 additional deaths and 569 new cases on Saturday.
The disease has proven most fatal among the elderly, but Illinois Governor J.B. Pritzker said on Saturday that an infant had died in his state.
