Conte might have been referring to Mourinho's charge down the touchline at Old Trafford when his Porto team eliminated United on its way to winning the Champions League in 2004.
"What has never happened to me and will never happen is to be suspended for match fixing".
Conte was asked whether it irritated him when rival managers talked about him or his players.
The Italian was even interpreted as suggesting that the United manager may be suffering from "dementia" after Mourinho had made a reference, 24 hours earlier, to his own behaviour and the fact that he no longer acts "like a clown".
Conte had reacted angrily this week to Mourinho's assertion that he did not need to act like a "clown" in his technical area, saying: "I think he has to see himself in the past - maybe he was speaking about himself in the past".
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"Maybe sometimes, I think that someone forgets what they said in the past or his behaviour and sometimes I think there is, I don't know the name, "demenza senile" when you are a bit... when you forget what you do in the past".
"I was asked about my passion and, you know, I was speaking about myself and then the question to the Chelsea manager was that I said he behaved like a clown", said Mourinho.
Conte received a four-month suspension while at Juventus in the 2012-13 season for failing to report match fixing at his previous club Siena.
"Everybody knows, I don't need the Chelsea manager to say, that I made mistakes in the past and I will make in the future, hopefully".
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Barkley is a signing driven primarily by Chelsea rather than Conte, but the Italian still spoke highly of the 24-year-old's ability and potential.
He said: "Look, I don't blame him".
Following last night's 2-0 triumph over Derby County, the Portuguese tactician conceded he wishes to end the minor squabble he commenced by adding managers shouldn't behave like clowns on the touchline. The question that comes to him comes to him wrong and because of that he has that out-of-control reaction.
After weeks of docile, drowsy press conferences in which he has largely borne the insults of other managers and the pressures of a faltering title defence with good grace, Antonio Conte finally snapped yesterday.
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