A league source tells ESPN NFL Insider Adam Schefter that Stafford has a second-degree separation and is planning to visit Dr. James Andrews as early as this week to determine how long he will miss.
How much time the second-year quarterback will be out will not be determined until after his visit with Andrews.
Stafford was knocked out of Detroit's season opener when Chicago's Julius Peppers sacked him Sunday. Stafford said after the game he would try to play in Week 2 at home against Philadelphia.
Stafford was very sore on Monday, but got good returns from tests, Schwartz said.
Shaun Hill started the third quarter in place of Stafford and appeared to make a game-winning throw to Calvin Johnson, but the catch was ruled incomplete when the ball popped up of the receiver's grasp when he hit the ground.
Information from The Associated Press was used in this report.