The two managers have been in embroiled in an insult-laden spat over the last 10 days.
Mourinho inferred Conte was “a clown” and brought up his “match-fixing” charge from his days in Italy.
While Conte has accused Mourinho of going senile and called him a “little man”.
On Friday, Mourinho said he now had only “contempt” for the Chelsea manager saying: “I think when a person insults another you can expect a response. Or you can expect contempt.
“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 the second time but now I change and now [I have] contempt. And for me contempt means the end of the story.”
But speaking after Chelsea’s goalless draw with Leicester, Conte hit back again and said he also feel contempt.
Conte said: ”I think that I said stop the story. But, it’s the same for me (contempt).
“I don’t know if he said this for me. I’m not worried. I sleep very well.”
Conte and Mourinho are not scheduled to meet until February 25 when Chelsea travel to Old Trafford.