Free agent Johnson visited the Jets on Wednesday, with ESPN reporting that the two parties wasted no time in finalising terms.
The Buffalo Bills, Miami Dolphins, New York Giants and Atlanta Falcons were all rumoured to be keen on Johnson, who is one of only six players to rush for at least 1,000 yards in each of his first six seasons in the NFL.
However, the 28-year-old posted a career-worst 3.9 yards per carry in 2013, prompting the Titans to part company with him earlier this month.
Johnson underwent knee surgery to repair a torn meniscus at the beginning of the off-season, and he passed a physical as part of his visit in New York on Tuesday.
Financial details of Johnson's contract have yet to be revealed.