Barton is said to have confronted Tykes boss Daniel Stendel in the tunnel following Saturday’s League One match, which ended in a 4-2 victory for the home side at Oakwell.
Footage appeared to show Barton being prevented from leaving Oakwell after the game, with police filmed blocking a car carrying the former midfielder.
South Yorkshire Police are investigating the incident.
“Barnsley Football Club can confirm that it has lodged a formal complaint to both the English Football League and the Football Association,” said a club statement. “This is following an alleged incident at the conclusion of the match on Saturday 13th April, between Joey Barton and Daniel Stendel.