Homes without heat would become uninhabitable as temperatures fell, state Governor Andrew Cuomo said.
New York City Mayor Michael Bloomberg put the figure at 30,000-40,000 people.
At least 106 US deaths – 40 of them in New York City – have been blamed on Superstorm Sandy, which struck on 29 October.
Residents who had so far refused to leave their homes would have no other option, Andrew Cuomo told a news conference on Sunday.
He also said there would be increasing pressure on public transport on Monday, as more people returned to work and the schools re-opened.
Fuel shortages were easing, but Andrew Cuomo urged New Yorkers not to hoard petrol, saying more supplies were on their way.
New York City opened warming shelters in areas without power and handed out blankets to residents who insisted on staying in homes without power.
But Michael Bloomberg urged those without heating to leave their homes if necessary.
“You can die from being cold. You can die from fires started when you use candles or stoves to heat your apartment,” he said.
“If you don’t know where to go, stop a cop on the street and say, please tell me where to go. They’ll help you. But we have to make sure that you are safe for a few days and that you have food and water for a few days.”
Temperatures fell to 39 F (4 C) on Sunday and are forecast to go as low as 30 F (-1C) on Monday.
About 730,000 people in New York state still do not have electricity, including more than 130,000 in New York City, the governor said.
Nearly a million people in the neighboring state of New Jersey remain without power, and petrol is being rationed.
Hundreds of runners who had been planning to take part in the New York marathon – cancelled by Mayor Michael Bloomberg on Friday – joined impromptu runs to raise funds or deliver aid.
The storm damage from Sandy is also affecting preparations for voting in Tuesday’s elections.
New Jersey residents displaced by Hurricane Sandy will be able to vote by email or fax, the state’s chief election official has decided.
They will be designated as “overseas voters” and can apply for mail-in ballots up until 17:00 on Election Day.
Michael Bloomberg said New York officials would do “anything we can” to help the board of elections, saying “they have real problems”.
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