The initial belief is the injury is "pretty bad," a source told ESPN's Jordan Raanan.
Jones was hurt while being tackled awkwardly on a third-quarter run. He returned to the game after a brief stint in the medical tent but was removed shortly after and replaced by Colt McCoy.
The second-year pro will undergo an MRI on Monday to determine the full extent of his injury, per Raanan.
With Sunday's win over the Cincinnati Bengals, the 4-7 Giants jumped past the Washington Football Team for the division lead in the NFC East.
Jones completed 63.5% of his passes for 2,122 yards and eight touchdowns against nine interceptions entering Week 12. He also totaled 384 yards and one score on the ground.
McCoy hadn't played a snap this season prior to the Bengals game. The 34-year-old spent the last five years in Washington, making 12 appearances, including seven starts. He previously started 21 games for the Cleveland Browns from 2010-12.