The 26-year-old free agent visited with the Giants last weekend and the team agreed to a deal on Friday pending Smith passes a physical, according to multiple reports.
Smith started one game for the Jets last season but suffered a torn ACL against the Baltimore Ravens in Week 7 on Oct. 23.
Johnson was the Giants' third-string quarterback last season until Manning's backup, Ryan Nassib, injured his elbow that eventually required surgery. Johnson, who will turn 31 in May, has not thrown an NFL regular-season pass since 2011 when he was a member of the Tampa Bay Buccaneers.
With the Giants expected to sign Smith, the Jets planned to bring in free agent quarterback Josh McCown for a visit Friday and Saturday, ESPN confirmed.
Smith was a second-round draft pick by the Jets out of West Virginia in 2013. He earned the starting job as a rookie and led the Jets to an 8-8 record.
The Jets went 3-10 in games Smith started in 2014 and he lost the starting job after being punched by teammate I.K. Enemkpali in training camp in 2015.
Smith continues to recover from his ACL injury and might not be able to participate in football activities until training camp.
Smith played in 33 games (30 starts) in his four seasons with the Jets, completing 57.9 percent of his passes for 5,962 yards with 28 touchdowns with 36 interceptions.