According to Pro Football Talk, the Cardinals scout was waiting to bring an unnamed player from his interview with the Falcons to one with Arizona. When Atlanta's time with the player expired, Cox confronted the scout verbally and then shoved the scout.
The altercation didn't escalate from there, according to Pro Football Talk.
"I intend to apologize to the young man for the incident that happened last night during the interview process," Cox, 48, said in a statement. "It shouldn't have happened."
The Cardinals have yet to comment on the incident.
Cox was known as a fiery player during his 12 seasons as a linebacker in the NFL. He retired after 2002, and started coaching several years later.
This was his second season with the Falcons after stints with the New York Jets, Tampa Bay Buccaneers, Miami Dolphins and Cleveland Browns. He was notably involved in a sideline scuffle with Falcons defensive tackle Ra'Shede Hageman in December.
As a player, Cox took the field for five teams after being drafted by the Dolphins in the fifth round in 1991. He was a three-time Pro Bowler and had 51.5 career sacks.