Johnson and the Jets reached agreement on a deal worth $72.5 million over five years, including $34 million in guaranteed salary, according to the NFL Network's Ian Rapoport.
Deals cannot be officially signed until the new league year begins on Wednesday at 4 p.m. ET.
Johnson has spent all six of his NFL seasons with the Rams, who used the franchise tag on the star cornerback in each of the past two years.
The 28-year-old Johnson had 65 tackles and a pair of interceptions last season for the Rams, who used their franchise tag this season on safety Lamarcus Joyner to finally give Johnson the chance to hit free agency.
A third-round draft pick of the Rams in 2012, Johnson has appeared in 85 games, making 62 starts, for the Rams. He has 18 interceptions, including a career-best seven in 2015.