Sanchez will seek a second opinion on his shoulder this week from renowned surgeon Dr. James Andrews, according to the source.
If Sanchez does have a tear in his labrum, he will not necessarily need surgery.
Sanchez, who suffered the injury last month in the Jets' preseason game against the New York Giants, was inactive for Sunday's season-opening victory against the Tampa Bay Buccaneers. Rookie Geno Smith started at quarterback.
Jets coach Rex Ryan declined to address reports of the injury after the game.
"He remains day to day for us, and that's what he'll continue to be," Ryan said.
Sanchez, who did not address the media after the game, was seen throwing with his left hand during pregame warm-ups Sunday. He is expected to miss at least a few weeks with the shoulder injury and could be placed on injured reserve, sources told ESPN.com.
Jets general manager John Idzik has downplayed the possibility of Sanchez being placed on season-ending IR, referring to the quarterback's return timetable as "unpredictable" earlier this week.
There also has been speculation that the Jets could release Sanchez, who is owed $8.25 million in guaranteed salary this season. Jets owner Woody Johnson dismissed that notion, however, stating that Sanchez "will be on the team."
ESPN.com's Rich Cimini contributed to this report.