Berry was injured in last week's 24-17 loss to the Washington Redskins. He left the team's temporary headquarters in the D.C. area -- where the team is staying this week preparing for the Jets game -- to return to Phoenix.
Coach Ken Whisenhunt was uncertain how long Berry will be sidelined.
"Hopefully, we'll get him back quickly," Whisenhunt said. "We'll know after this week a little better as far as how it responds. Because of the way he responded in going back in the game on Sunday, we were hopeful it wasn't bad.
"But when they did the MRI, it found out that he did have a tear in there. Now, it's see how he reacts to it and see how he heals."
The 11th-year lineman had a sack in each of Arizona's first three games.
Whisenhunt said the trio of Travis LaBoy, Chike Okeafor and Clark Haggans would likely rotate at the two end spots against the Jets.
The Cardinals, who are staying at a northern Virginia hotel and practicing at Catholic University in Washington, D.C., this week, will depart for New York via train on Saturday.