Mourinho reportedly felt Conte's behaviour was humiliating after Chelsea inflicted a bruising 0-4 defeat on Manchester United here on Sunday. [EPL game week 9: Roundup, team of the week]
Goals from Pedro Rodriguez, Gary Cahill, Eden Hazard and N'Golo Kante sealed a thumping victory.
The match was Mourinho's first at Chelsea's home ground Stamford Bridge after his sacking by Chelsea midway last season.
The Portuguese exchanged words with Conte after full time, and according to several reports, he was unhappy with the conduct of Conte at the end of the match.
Sky Sports Italia quoted Mourinho as saying to Conte: "Don't celebrate like that. At 1-0 ok but not 4-0 - that's humiliating to us."
However, when Mourinho was asked about the exchange, he refused to elaborate.
"It was fine, I was not expecting anything different here. The crowd behaved in a normal way. My words were with Antonio Conte were for him, not for you," SkySports quoted Mourinho as saying.
Conte dismissed the uproar as trivial.
"We were 4-0 up and our crowd was flat. I wanted our supporters to applaud our team because they deserved it," said Conte.
"I think that today, it was right to call our fans in a moment I was listening only to the supporters of Manchester United at 4-0."
"So I called the fans to give the players a great clap after this kind of performance. The players, after a 4-0 win, deserved a great clap. It's very normal," the Italian added.
The acrimony continued in the ground tunnel after a member of Mourinho's backroom staff reportedly refused to shake Conte's hand and was confronted by one of the Italian's assistants.