Winston suffered the injury in Tampa Bay's 38-33 loss to the Arizona Cardinals Sunday after being hit by Chandler Jones on the first possession of the game
"I've come out of a game before, but I don't think I've ever missed a game [on any level]," Winston said Sunday, per Jenna Laine of ESPN.com. "I don't think I'm going to miss a game, either."
Winston was able to return to Sunday's contest after initially suffering the injury but eventually left the game for good in the second quarter, finishing 5-of-10 for 61 yards.
"Jameis is as tough as they come," head coach Dirk Koetter said, per Laine. "This isn't about toughness. This is about, 'He's got a shoulder injury and couldn't throw the football. He couldn't do his job.' We just have to see what happens."
Winston's backup, Ryan Fitzpatrick, played well in the loss, throwing for 290 yards, three touchdowns and two interceptions while nearly leading the team back from a 31-0 deficit early in the third quarter. Still, after starting 2-3, losing Winston for any time would be a major blow for a Buccaneers team looking to stay competitive in a tight NFC South.