The Jets also have a deal in place with Teddy Bridgewater, according to CBS Sports' Bryant McFadden.
McCown signed a one-year, $6 million deal with Gang Green prior to the 2017 season to serve as a veteran stopgap, and he thrived in that role as Robby Anderson, Jermaine Kearse and Austin Seferian-Jenkins all emerged as legitimate weapons in a surprising receiving corps.
By season's end, McCown set career highs in completion percentage (67.3), passing yards (2,926) and passing touchdowns (18) at 38 years old.
"I think Josh did a very good job," Jets general manager Mike Maccagnan told reporters after the season, per NJ.com's Darryl Slater. "I still think there is good football left in Josh. I think from an intangible standpoint, he was outstanding for our building, our locker room.
"Based on what Josh has done, I would have no problem bringing him back. But of course, that's another thing that's going to play itself out here over time. Everything from our standpoint has been exceptionally positive with Josh."
Former Jets offensive coordinator John Morton echoed that sentiment.
"He's been consistent," Morton said, according to the New York Daily News' Daniel Popper. "He's perfect for this offense.
Morton added: "He is just another coach. When you have someone like that in your room, it's an awesome feeling."
And therein lies the main reason McCown made sense for the Jets.
While he's still capable of leading the offense as a starter if need be, McCown is also the ideal mentor for whichever quarterback the franchise pegs at its signal-caller of the future, whether that be Bridgewater or somebody the Jets select in the NFL draft.