Clark exited Sunday's game in the second quarter, and was wearing a walking boot on the sideline.
Brody Eldridge, another tight end, left Sunday's game with a hand injury in the first quarter.
Colts head coach Jim Caldwell is unsure how long Clark and Eldridge will be out, but he stated that both players suffered "pretty significant injuries."
"What I mean by that is more than a week," Caldwell said.
If the Colts are left without Eldridge and Clark for a few weeks, the only tight end Indianapolis has left on the roster is Jacob Tamme.
"We do have to do something in those areas and we're working on that as we speak," Caldwell said.
The 32-year-old Clark has 28 receptions for 287 yards and two touchdowns in nine games played this season, while Eldridge has six catches for 33 yards.