Prescott has been held out of team drills since late July because of a latissimus strain in his right shoulder. He has had no setbacks and remains on track with a progressive throwing program designed to increase volume of work, the source said.
On Wednesday, coach Mike McCarthy said "there's a good chance" Prescott won't play in the preseason. Heading into training camp, the team had hoped to get Prescott plenty of reps now that he has recovered from the dislocation and compound fracture of his right ankle that ended his 2020 season in Week 5.
Dallas opens the regular season on Sept. 9 with a Thursday night game against the defending champion Tampa Bay Buccaneers.
NFL Network first reported on Prescott's availability for Week 1.
Information from ESPN's Todd Archer was used in this report.