Peterson's injury does not require surgery but could require multiple weeks of rest to heal a source told ESPN's Josina Anderson.
When asked about Peterson's long-term prognosis, Arians said, "I have no clue. It's something that he and doctors are working through."
"He's walking around fine. It's just a matter of being able to take a hit and getting all of his strength back," Arians said.
Arizona hosts the Tennessee Titans this Sunday.
Peterson did not practice Thursday after missing Sunday's loss to the Detroit Lions.
He has rushed for 448 yards on 129 carries for the Cardinals since joining Arizona in a midseason trade from the New Orleans Saints.