Leary was penciled in as the starting left guard, but he will now likely need two to four weeks to recover from the procedure, according to a source.
Meanwhile, Nate Livings is at least two weeks away from returning to the field after undergoing surgery on his right knee earlier in training camp.
The Cowboys had an agreement with Brandon Moore on a one-year deal last week, but the veteran elected to retire instead. The Cowboys have an offer on the table for six-time Pro Bowler Brian Waters, who has not played since 2011 with New England.
Leary missed the first week of training camp because of a calf injury, but he started and played in most of the Cowboys' first two preseason games. He spent most of last year on the practice squad after the Cowboys signed him as an undrafted free agent and guaranteed him more than $200,000.
With Leary and Livings out, David Arkin will start at left guard in Saturday's preseason game at Arizona.