The Manchester United manager has traded insults with his Chelsea counterpart over the past two weeks, with their dispute becoming increasingly bitter.
"I think when a person insults another you can expect a response. Or you can expect contempt," Mourinho told reporters.
"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 (I have) contempt. And for me contempt means the end of the story."
The enmity between the two men stretches back to early last season, with Mourinho accusing Conte of humiliating him with the scale of his celebrations as United fell to a 4-0 defeat at Stamford Bridge in October 2016.
The two managers have continued to snipe at each other since then, with Ander Herrera sent off during a bad-tempered FA Cup quarter-final in London last March, as Conte accused United of trying to stop Eden Hazard by fouling him repeatedly.
Conte has also criticised United's spending, while Mourinho even mocked the Chelsea head coach for having a hair transplant.
However, their comments have acquired more bite lately, beginning when Mourinho implied that Conte was a "clown" for his dramatic touchline celebrations, although he later claimed he was not referring specifically to the Italian.
Conte then accused the United manager of going senile, with Mourinho responding by referring to the four-month suspension that Chelsea's manager received in 2012 for failing to report match-fixing while at Siena.
Conte, who was cleared by a court of any wrongdoing in 2016, retorted by calling Mourinho a "little man".
- Sanchez is 'phenomenal' -
Meanwhile, Mourinho, whose side host Stoke in the Premier League on Monday, has refused to rule out a January move for Alexis Sanchez, describing the Arsenal forward as a "phenomenal" player.
But the United manager does not want to be drawn on his chances of beating Manchester City to a deal, saying it is not "ethical" to talk about another club's player in such a way.
City are favourites to sign Sanchez, who is out of contract at the end of this season and has made clear he has no intention of staying at Arsenal.
Mourinho, whose squad has just returned from a warm-weather training camp in Dubai, said: "I don't know if it's ethical or correct to be speaking about players of other clubs.
"If somebody speaks of one of my players to tell we are interested, I wouldn't be very happy.
"Sanchez is an Arsenal player. Probably this weekend he is going to defend Arsenal colours so I don't think it's correct to say things about Alexis Sanchez.
"At the general level, what I can say is myself and the Manchester United board, and the owners, we don't believe a lot in the January market.
"But in relation to Alexis Sanchez, I don't say a word. The only word I can say is he is a phenomenal player. Apart from that, he is an Arsenal player and I feel I shouldn't say much more than this."