Jones said during an interview with 105.3 The Fan on Tuesday that Prescott won't be placed on injured reserve because he could return from his broken thumb within four weeks.
"We want him to be in consideration for playing within the next four games," Jones said
Prescott was originally expected to be sidelined for six to eight weeks after suffering a fracture near his right thumb in the fourth quarter of Sunday's 19-3 loss to the Tampa Bay Buccaneers.
If placed on IR, Prescott would have to miss a minimum of four games, however, after Monday's surgery, Jones is optimistic the 29-year-old won't be out that long.
"We feel better about it than we did Sunday night," Jones said.
With Jones' new timeline, Prescott could be back playing in Week 5 against the Los Angeles Rams on October 9.
"Dak has a real chance to be back out there throwing the ball pretty quick," Jones said.
The Cowboys host the 0-1 Cincinnati Bengals on Sunday with Cooper Rush set to start at quarterback.
The 28-year-old Rush made his lone career start last season in Week 8 at Minnesota with Prescott nursing a calf strain and threw a winning touchdown pass to Amari Cooper in the final minute of a 20-16 victory.
Rush threw for 325 yards in that game and was seven of 13 passing for 64 yards Sunday after Prescott got hurt.
Prescott struggled against Tampa Bay prior to his injury, completing 14 of 29 passes for 134 yards with one interception, compiling a 47.2 passer rating – the lowest by any QB in Week 1.