McCown, 39, has been contemplating retirement and will become a free agent this week.
"While I still have a strong desire to continue playing, my family dynamics at this point are such that I would like to take more time before making the commitment to play in 2019," McCown's statement read. "I have spoken to the Jets to make them aware. I'm excited about spending time with my family as we process what's next."
McCown, who will turn 40 on July 4, said at the conclusion of the Jets' season that he would discuss his future with his family. His body feels fresh, as he got on the field for only four games -- three starts, plus a one-series cameo.
After enjoying a career year in 2017, McCown lost his job in the preseason to first-round pick Sam Darnold. It wasn't much of a competition. McCown spent nearly the entire preseason watching from the sideline as Darnold was given a clear path to the starting job.
McCown embraced his role as Darnold's mentor.
When McCown re-signed last March (one year, $10 million), he knew there was a good chance the team would draft a quarterback in the first round. The Jets wanted him back because of his unselfish attitude and willingness to help young quarterbacks. It was costly from a financial standpoint, but it was "money well spent," Jets CEO Christopher Johnson said.
The well-traveled McCown was pressed into starting duty for three straight games when Darnold was out with a foot injury in Weeks 10 to 12 but wasn't able to recapture his 2017 form. McCown struggled, completing only 54.5 percent of his passes for 539 yards, with one touchdown and four interceptions. The Jets lost all three games.
The Jets have plans to look elsewhere for a backup quarterback. One possibility is pending free agent Brock Osweiler, who played under Adam Gase with the Miami Dolphins and Denver Broncos. Gase also could look to reunite with Ryan Tannehill, if he's released by the Dolphins.
Aside from Darnold, the only other quarterback on the Jets' roster is former New York Giants draft pick Davis Webb, but he has no regular-season experience. He spent most of last season on the Jets' practice squad.
McCown never has been a star player in the league, but his career is remarkable in its own right. He has played for seven teams over 15 seasons, beginning with the Arizona Cardinals in 2002. In fact, he is the lone remaining player from the 2002 draft still active.
McCown signed with the Jets in 2017 and landed the starting job by default, as young quarterbacks Christian Hackenberg and Bryce Petty struggled in the preseason. McCown wound up setting career highs for completions (267), yards (2,926) and touchdown passes (18). The players voted him as the team MVP even though he missed the last three games with a broken hand.
He has spent most of his career as the proverbial "bridge" quarterback, bouncing from one rebuilding situation to another. His career record as a starter is only 23-53. For his career, McCown has 17,707 passing yards, 98 touchdown passes, 82 interceptions and a 79.7 passer rating.
ESPN's Rich Cimini contributed to this report.