Cousins is likely to hit the open market after registering three consecutive 4,000-yard passing seasons with the Washington Redskins.
The 29-year-old Cousins played under the franchise tag each of the past two seasons, and while the Redskins could tag him again, it is unlikely.
Washington already agreed to a deal with the Kansas City Chiefs to acquire quarterback Alex Smith, meaning Cousins likely isn't in its plans.
Cousins could be in line for the richest quarterback contract in NFL history, and the Jets are among a handful of teams that would benefit greatly from his arrival.
During his three full seasons as the Redskins' starting quarterback, Cousins received one Pro Bowl nod and led the team to one NFC East title.
In 2017, Cousins completed 64.3 percent of his passes for 4,093 yards, 27 touchdowns and 13 interceptions, while rushing for an additional four scores.
The Jets went 5-11 in 2017 as 38-year-old journeyman Josh McCown started 13 games at quarterback.
McCown was serviceable with 18 touchdowns and nine interceptions, but New York is clearly in need of a long-term answer under center.
If the Jets are unable to sign Cousins, then they may be compelled to use the No. 6 overall pick in the 2018 NFL draft on a quarterback such as Sam Darnold, Josh Rosen, Baker Mayfield or Josh Allen.