The deal is worth $1.5 million, with a maximum value of $4.5 million with incentives, Adam Schefter of ESPN reports.
The Jets have sought a veteran quarterback in recent weeks and had interest in Andy Dalton before he signed with the Cowboys.
Flacco joins a quarterbacks room with starter Sam Darnold and backups David Fales, Mike White and rookie James Morgan. Fales has no career starts and has played only five games with 48 attempts, and White has never appeared in a regular-season game.
Flacco, who spent last season in Denver after 11 years in Baltimore, had neck surgery in early April. He is not expected to be cleared until somewhere around late August.
Flacco, 35, has a 98-73 record in his 12 seasons, throwing for 40,067 yards with 218 touchdowns and 141 interceptions.