The pair clashed in a defensive wall during Juve's 1-0 win.
"It was a manly game with some determined challenges from both sides," Chiellini told Mediaset Premium.
"I apologised to Pjanic for the contact, but there are no problems between us. In games like this I've given a few knocks and received quite a few as well.
"I wasn't trying to elbow him, as I was just trying to block him from moving out of the wall, the same way you do in basketball. We were jostling each other, the referee saw and evaluated everything.
"You can't say that just glancing by someone is the same as punching them in the face. People are trying to build up controversy that doesn't exist."
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