Reddick sustained the injury in Saturday's win over the Angels and is not expected to do any baseball activity until the A's return home next week.
The good news is there's no structural damage to the knee.
"But we're concerned about it to the point where he's not doing anything for 10 days," said manager Bob Melvin. "It's something we want to be careful with."
Reddick was batting just .214 with four home runs and 22 RBIs through 50 games with the A's, who added a left-handed bat Sunday to offset the loss of Reddick by recalling catcher Stephen Vogt from Triple-A Sacramento.
Craig Gentry got the start in right field in New York on Tuesday.
Jane Lee is a reporter for MLB.com. Read her blog, Major Lee-ague, and follow her on Twitter @JaneMLB. This story was not subject to the approval of Major League Baseball or its clubs.