The franchise is owned jointly by Manchester City and the New York Yankees.
Villa won the Spanish league title with Atletico Madrid this season and is part of the Spain squad which will try to retain the World Cup this summer. However, the 32-year-old is expected to move to the US afterwards.
"It's what I've tried to do in my career, to overcome challenges, and I think this is another big challenge," said Villa. Frank Lampard is also rumoured to be joining New York City this summer.