The 23-year-old player was to undergo surgery Monday, the newspaper said.
Gronkowski, accompanied by Thomas Gill, the team doctor, left Sunday's 59-24 Pats victory over Indianapolis in the fourth quarter with his arm hanging limply at his side.
It was unclear how long Gronkowski will be out, the league source said.
Gronkowski had seven catches for 137 yards and two touchdowns in the game. He was injured while blocking during the game's final extra point.