Mr Barry, it was claimed by Chelsea boss Jose Mourinho, was influenced by Alex Ferguson at half time.
"Maybe when I turn 60 and have been managing in the same league for 20 years and have the respect of everybody I will have the power to speak to people and make them tremble a little bit," Mourinho said.
"The referee controlled the game in one way during the first half but in the second they had dozens of free-kicks. It was fault after fault, dive after dive.
"But I know the referee didn't walk to the dressing rooms alone at half-time. He should only have had his two assistants and the fourth official with him, but there was also someone else."
Mourinho has been told he will face no action from the FA for his comments.
Mr Barry last officiated a Cardiff game in 2003. He was at Ninian Park when City beat Derby 4 - 1. His two assistants at Elland Rd will be Roy Burton from Derbyshire and Trevor Massey from Cheshire.