Hayward, 33, will have to miss at least four weeks, but the Falcons are worried it might be longer. Coach Arthur Smith said Monday the injury could be "long-term."
Hayward sustained the injury in the second half of Sunday's win over the San Francisco 49ers. He left the game and did not return.
Starting CB A.J. Terrell (thigh) also left the game and didn't return, but that injury doesn't appear as serious.
Tight end MyCole Pruitt took Hayward's spot on the active roster.
Hayward has 17 tackles and one interception in six starts this season, his first in Atlanta.