The two Premier League bosses have traded barbs since the new year after Mourinho's comment he did not feel the need to behave like "a clown" during games touched a nerve with Conte.
That sparked an extraordinary war of words between the pair, where Conte accused Mourinho of having "demenza senile", which directly translates as senile dementia.
It prompted the United boss to hit back by appearing to allude to a four-month suspension Conte served in relation to match-fixing.
Conte, who was later cleared of any wrongdoing on that front, responded by accusing Mourinho of being "fake" and "a little man".
On Friday, Mourinho conducted his first press conference since one of his Chelsea successor's latest remarks.
"I think when a person insults another, you can expect a response or you can expect contempt, silence," Mourinho said.
"The first time he insulted me I had a response, a response that I know that touched the point where he really feels hurt.
"Then he insulted me for a second time, but now I change, and now (there is) contempt and for me, contempt means end of the story."