Both managers were sent to the stands after the incident towards the end of Everton's 2-0 win at Manchester City where they had to be separated after Moyes had caught the ball and Mancini thought he was time wasting so he went to confront him.
The pair were pulled apart by official Howard Webb and were sent to the stands and they will now wait to see what, if any, action the FA will take.
Moyes believes he had done nothing wrong and maintains he should not have been sent off. He said: "Next time I might just let the ball run through.
"I caught it because I was making a substitution, I would have thought it would actually have helped him.
"I was very surprised, I didn't know what I had done wrong.
"Maybe I held onto the ball a second or two longer than I should have done but I was trying to make a substitution.
"But if I had let the ball go, let it run down the touchline - I actually speeded it up."
"I wasn't the one waving my hand for bookings. "I shouldn't have been sent off, simple as."