Cuomo said he spoke with Yankees owner Hal Steinbrenner and President Randy Levine and Mets Chief Operating Officer Jeff Wilpon who informed him of the plan.
"The state is going to review their protocols in place to make sure they're doing it safely and appropriately," Cuomo said.
The decision comes amid a surge of positive COVID-19 tests in Florida.
The state reported 4,049 new cases on Saturday, breaking its own record for the third day in a row.
Cuomo said one of the reasons for the decision is that New York's coronavirus infection rate is one of the lowest in the country.
The teams will train at their respective home ballparks, the Yankees at Yankee Stadium and the Mets at Citi Field.
Cuomo said the Mets will start a soft opening this week.
"To have spring training in New York, that is a really great development," Cuomo said. "It's where I think they should always be."
While it's exciting news for New York, it may be moot if Major League Baseball and the MLB Players Association can't come to an agreement on how to start the season.
The two sides have been negotiating ever since spring training was shut down due to the coronavirus pandemic.
Right now, they're at a stalemate as the League said it won't counter the MLBPA's latest offer to play 70 games this season.
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