"We want him to be a consideration for playing for us within the next four games," Jones said.
Prescott underwent surgery on his right thumb after experiencing an injury during Sunday's 19-3 loss to the Tampa Bay Buccaneers in Week 1 of the 2022 NFL season.
Sources initially told ESPN that Prescott was expected to be out six to eight weeks, however, Jones believes the quarterback "has a real chance to be back out there throwing the ball pretty quick."
Prescott exited the game during the fourth quarter after an apparent injury to his throwing hand, which was determined to be a fractured thumb that will require surgery.
Multiple sources told ESPN that Prescott will likely miss 6-8 weeks of action and is expected to be placed on injured reserve.
The former Mississippi State standout was replaced by backup Cooper Rush, who is expected to start for the Cowboys during their Week 2 game against the Cincinnati Bengals.
Prescott finished Sunday's game with 134 yards and one interception on 14 of 29 passing and was sacked twice by the Buccaneers defense.
Sunday's loss marked the first time Dallas was held without a touchdown in a season opener since 2001 and the first time the Cowboys had scored less than 6 points in an opener since being shutout by the New Orleans Saints in 1989, which was the first game of Jerry Jones' tenure as owner, as well as the debuts of Pro Football Hall of Fame quarterback Troy Aikman and head coach Jimmy Johnson.
Prescott was reported to be limited during Thursday's (September 8) practice due to a sore ankle, claiming a new pair of cleats caused the issue but insisted he was fine to play in Sunday's game without any issue and completed full workouts on Friday (September 9) and Saturday (September 10).
The former NFL Offensive Rookie of the Year suffered a compound fracture and dislocation of his right ankle during the fifth game of the 2020 season before returning to start 16 of the Cowboys' 17 games in 2021, missing only one start due to a calf strain and setting a franchise single-season record of 37 passing touchdowns.