Mourinho and Conte have engaged in a war of words in recent weeks that has seen both men fire increasingly venomous barbs.
And speaking on Friday, Mourinho showed no inclination to let tempers calm as he once more went on the warpath.
"I think when a person insults another you can expect a response," the United manager fired to reporters in a press conference.
"Or you can expect contempt, silence. 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 contempt and for me contempt means end of the story."
Chelsea got the better of the Red Devils in their first Premier League meeting of 2017-18, winning 1-0 at Stamford Bridge.
The teams will meet again in the return fixture at Old Trafford in February, with fireworks expected if the managers continue to clash in their bitter off-field spat.