Clarets player of the year Barton, 33, was one of the last players to arrive at Burnley Town Hall as the players were about to lift the trophy.
“The Football League advised the club that 25 medals were to be presented, but the club put forward 27 players,” Burnley tweeted.
Defender James Tarkowski also missed out on receiving a medal.
The Clarets secured an immediate return to the Premier League by claiming the Championship title.
When the trophy was passed around the squad, Barton refused to lift it, instead holding aloft an inflatable one he got from the crowd.
“No medal? What’s going on?” the former Manchester City and Newcastle midfielder tweeted.
The club wrote: “Burnley Football Club would like to apologise for the mix-up during the medal ceremony.
“We would like to apologise to the Football League for this administrative error and every player will be receiving one.”
Tarkowski added on Twitter: “It was only banter, as if I’d refuse a Championship winning medal! The ribbon wouldn’t fit over my ears so waiting for one with more give!”
Credits: BBC Sport