The 33-year-old, who will compete with Geno Smith for the role of starting quarterback next season, joined the MetLife Stadium side shortly after the Jets had confirmed that Mark Sanchez had been released.
"Maybe I can come here and be an asset whenever my number's called," said Vick. "I'm excited, I'm elated, and I feel like this is a great opportunity for me and my family."
Jets offensive coordinator Marty Mornhinweg added: "We are excited to acquire Michael Vick.
"He's a great talent who's won many games, including playoff games, in this league. He has the respect of both his teammates and opponents.
"He's here in a role where he's going to compete and push Geno Smith to become the very best that he can be. Michael will be able to go in and play at a high level and a winning level if called upon."
Mornhinweg and Vick previously worked together at the Philadelphia Eagles.