The plant is near the Ramapo Fault in New York, where substantial earthquakes are infrequent.
"We are going to check into it ... immediately," Cuomo said Wednesday.
Cuomo noted news reports following the nuclear plant crisis in Japan that stated Indian Point is one of the most vulnerable plants in the country because it is near a geological fault line and close to a metropolis.
"This plant in this proximity to New York City was never a good risk," the Democrat said. "But this is new information we are going to pursue."
Cuomo said the issue was discussed with the leaders of the Senate and Assembly in a closed-door meeting Wednesday. The plant is a major power producer for New York City.
A spokesman for Indian Point didn't immediately respond to a request for comment.
The federal Nuclear Regulatory Commission is reviewing a renewal application for Indian Point. Cuomo opposed the effort when he was state attorney general.
According to new risk estimates by the U.S. Nuclear Regulatory Commission provided to msnbc.com, Indian Point ranks no. 1 on a list of U.S. nuclear reactors at risk for catastrophic failure due to an earthquake.
The NRC has calculated the odds of such an event. At the typical nuclear reactor in the U.S., there's a 1 in 74,176 chance that the core could be damaged by an earthquake, exposing the public to radiation, according to msnbc.
That's 10 times more likely than you winning $10,000 by buying a ticket in the Powerball multistate lottery, where the chance is 1 in 723,145.
The chance of a core damage from a quake at Indian Point 3 is estimated at 1 in 10,000 each year.
Under NRC guidelines, that's right on the verge of requiring "immediate concern regarding adequate protection" of the public, msnbc.com said.
Entergy and the NRC dispute msnbc.com's story. They said nuclear reactors are never ranked, only evaluated.
The two reactors at Indian Point generate up to one-third of the electricity for New York City. The second reactor, Indian Point 2, doesn't rate as risky, with 1 chance in 30,303 each year.