Smith suffered the injury during Sunday's game against the Baltimore Ravens when he twisted his knee on a second-quarter sack. Making his first start since Dec. 28, 2014, Smith completed 4 of 8 passes for 95 yards and a touchdown before leaving the game.
As a result of Smith's injury, the Jets will return to Ryan Fitzpatrick as their starting quarterback.
Smith, the Jets' 2013 second-round pick, may have played his final game with the Jets. He will be a free agent after the season.
"Something just felt funny when I got up," Smith said of the injury after Sunday's game. "I don't think it will be anything too significant, but we'll see.
Fitzpatrick, who was demoted last week, came off the bench to replace Smith and led the Jets to a 24-16 win over the Ravens to end their four-game losing streak.
Fitzpatrick, who enjoyed a career year in 2015 and landed a one-year, $12 million contract, was benched following a four-game losing streak. He has thrown a league-high 11 interceptions.
"The biggest thing in this game, in order to last, is to have belief in yourself, because when the owner stops believing in you, the GM stops believing in you, and the coaches stop believing in you, sometimes all you have is yourself," said Fitzpatrick, who was benched last week. "That's kind of something I've had to deal with before, something I'm dealing with now."
Bowles did not have a problem with Fitzpatrick's comments after the game.
"I know Fitz, and I know what he means," Bowles said Monday. "I have no problem with him venting his frustrations."
Second-year quarterback Bryce Petty, who has no regular-season experience, will serve as the backup while rookie Christian Hackenberg will be No. 3 on the depth chart as the Jets prepare this week for the Cleveland Browns.