According to Manish Mehta of the New York Daily News, Falk is dealing with a hip injury, but he was not listed on the injury report before or after New York's Week 5 loss to the Philadelphia Eagles, which marked his final appearance for the team.
Falk is the second player to file an injury grievance against the Jets this season, as guard Kelechi Osemele filed one last month.
Per ESPN's Rich Cimini, Osemele's agent, Andrew Kessler, said a grievance was filed in an effort to recoup losses from the Jets fining Osemele for conduct detrimental to the team when he refused to practice and underwent unauthorized shoulder surgery. Osemele was subsequently released by the team.
When Jets quarterback Sam Darnold was out due to mononucleosis, Falk made two starts for the team. He also played most of the Jets' Week 2 loss to the Cleveland Browns after Trevor Siemian suffered an ankle injury.
The 24-year-old Falk went 0-2 as a starter, and during his three appearances, he completed 64.4 percent of his passes for 416 yards, no touchdowns and three interceptions.
New York's 31-6 loss to Philadelphia was an especially tough game for Falk, as he threw two interceptions, fumbled twice and was sacked nine times.
When Darnold was officially cleared to return in Week 6, the Jets released Falk without an injury designation and decided to go with longtime Adam Gase disciple David Fales as the backup.
Even with Darnold back in the fold, the Jets have struggled mightily this season, and they are among the NFL's worst teams with a 1-7 record after falling to the previously winless Miami Dolphins last week.