Mourinho exchanged heated words with Conte at the final whistle at Stamford Bridge, with Sky Sports reporting that the Portuguese coach was unhappy with Conte's attempts to encourage the home fans to make more noise with the score at 4-0.
The 53-year-old is reported to have told Conte in Italian: "You don't celebrate like that at 4-0. You can do it at 1-0, otherwise it's humiliating for us."
Speaking after his team's comprehensive win, Conte refused to divulge what Mourinho told him at the final whistle but insisted that his actions were not disrespectful.
"I was a football player, so I know how to behave on the pitch. I have great respect for everyone," he was reported as saying.
"There were United fans who kept singing, while ours were in silence after a great performance from my team. I wanted to call the crowd to applaud the players, which seems normal enough to me.
"Nothing happened with Jose, especially as my behaviour seemed normal. I didn't make fun of anyone, as I always demand respect.
"If you want to cut the emotions, we can stay at home and I will change jobs. I did something absolutely normal. I didn't do anything to humiliate Mourinho."
Goals from Pedro, Gary Cahill, Eden Hazard and N'Golo Kante handed Mourinho's heaviest ever Premier League defeat on his first appearance back at Stamford Bridge since being sacked as Chelsea boss last December.
The defeat was also United's biggest in the league since losing 6-1 to Manchester City in 2011.
Mourinho declined to elaborate on his verbal exchange with Conte in the post-match press conference.
"You know me: I spoke to Conte, not to you," the United manager was reported as saying by the Guardian.
"I'm not the kind of guy to come here and share with you things I don't want to share. What is between me and Antonio is for us... of up to him to talk if he wants to. That is his problem. I have no comment."