Jones took 90 percent of the first-team snaps, per the report.
Jones was pulled after three series in the Patriots' loss to the Chicago Bears on Monday night. Patriots coach Bill Belichick said after the game that Jones wasn't removed for performance. However, he also said it wasn't a medical decision.
Rookie Bailey Zappe replaced Jones and played the rest of the game, throwing for 185 yards and a touchdown with two interceptions.
Belichick refused to dive into "hypotheticals" Tuesday morning about which quarterback would draw the start against the Jets.
Both Belichick and Jones said the plan all along was to go with both QBs against the Bears, a notion that reportedly was news to other members of the team.