Seferian-Jenkins, who is getting a second opinion, suffered the injury during the Bucs' 26-19 victory against New Orleans on Sunday.
The second-year player has caught seven passes for 139 yards and two touchdowns as the Bucs have gotten off to a 1-1 start this season.
Before the injury, Seferian-Jenkins had become an early favorite target for rookie quarterback Jameis Winston. Seferian-Jenkins was one of the few bright spots for the Bucs in a Week 1 blowout loss to the Tennessee Titans, when he posted 110 yards receiving on five catches with one touchdown.
His absence will strain the Bucs' depth at tight end. Brandon Myers, who had 22 catches for 190 yards with Tampa Bay last season, was listed as the No. 2 tight end to start the week and should take on a larger role without Seferian-Jenkins. Myers has yet to catch a pass this season.
Luke Stocker, who had seven catches for 41 yards last season, and Cameron Brate, signed by the Bucs off the New Orleans Saints' practice squad Tuesday per ESPN NFL Insider Adam Caplan, are the other tight ends on the roster.
Myers' best season came with the Oakland Raiders in 2012, when he had 79 catches for 806 yards and four touchdowns.
ESPN.com's Andrew Astleford contributed to this report.