Johnson agreed to terms with the Giants on Friday, according to his agent. He impressed the Giants last year, despite not throwing a regular-season pass. Smith, 26, could be viewed as a developmental project as Manning prepares for his 14th season.
With the Giants expected to sign Smith, the Jets are bringing in free-agent quarterback Josh McCown for a visit Friday and Saturday, a source confirmed to ESPN's Rich Cimini.
Smith, the Jets' second-round pick in 2013, is best remembered for his 2015 locker-room altercation with IK Enemkpali. It was a career-altering moment for Smith, who went from starter to forgotten backup in a split second.
Enemkpali, claiming Smith owed him $600, fractured the quarterback's jaw with a punch to the face. Smith required surgery and missed nearly two months, losing his job to Ryan Fitzpatrick, who enjoyed a career year.
"Every day I'm pissed off until I get my job back," Smith told ESPN during last summer's training camp. "Until I'm a starting quarterback, I'm pissed off every day. That's my mentality, that's my competitive nature. I want to win so badly, deep inside of me. I'm not pissed off at anyone, but I do believe I'm a starting quarterback in this league, and I believe I can do great things."
Smith got another shot last season, but it was short-lived. Starting for the demoted Fitzpatrick in Week 7, Smith tore knee ligaments in the second quarter and was finished for the season. He underwent surgery in November, vowing to be ready for training camp.
Smith, who started as a rookie, compiled a 12-18 record in four seasons with the Jets. He struggled with turnovers, as he threw 36 interceptions -- one every 24 pass attempts.
It was an eventful run, not all of it positive. Smith was fined by the NFL for cursing at a fan at a home game and he received a torrent of criticism for missing a team meeting on a West Coast trip, claiming he got the time zones mixed up.