Smith was hurt while blocking on running back Derrick Henry's historic 99-yard touchdown run.
Taking over for injured starter Delaney Walker, Smith has 20 catches for 258 yards and three touchdowns this season.
Having caught passes in seven straight games, Smith didn't see any passes come his way in Thursday's 30-9 victory.
Right tackle Jack Conklin is also expected to miss time after suffering a knee injury against the Jaguars, according to NFL insider Ian Rapoport.
--Field Level Media
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