Neither manager would reveal the contents of the conversation but Sky Italia said Mourinho 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."
Conte gestured to the crowd to turn up the noise levels after N'Golo Kante scored Chelsea's fourth goal.
British media reported that the ill-feeling spilled into the tunnel where a member of Mourinho's staff refused a handshake from Conte, and others were involved in verbal disputes.
Chelsea's manager denied that he had been trying to humiliate United.
"There has always been respect for all, and naturally for Manchester United," said Conte. "Nothing happened. I did something absolutely normal. I didn't do anything to humiliate Mourinho.
"The players, after a 4-0 win, deserved a great clap. It's very normal. We live with emotions. If you want to cut the emotions, we can stay at home and I will change jobs."
Since being beaten 3-0 by Arsenal last month, Chelsea have won three straight league games, scoring nine goals without conceding and now sit a point behind the leaders.
Conte's decision to go with a 3-4-3 has reinvigorated the midfield, where Nemanja Matic and Eden Hazard enjoyed standout performances against United. Hazard's goal took his Premier League tally to four, equalling his total for the whole of last season.
Matic said the Arsenal defeat had been the turning point in Chelsea's transformation.
"I think the Arsenal game was a wake-up call for us," he told the Evening Standard. "We knew that we had to do better, to work more and that's what we have done. It's easier to say you need to work hard, than to actually do it.
"But after the Arsenal game we needed to improve. We had quality in the team, but we had to show it. In the last three games we have shown quality, scored goals and not conceded, which is very important.
"It is not easy to win 4-0 against a strong team like Manchester United. This was the real Chelsea."
Matic said he had respect for his former coach Mourinho. "He was our coach and we respect him, me especially. He brought me back here from Benfica so I have big respect for him."
(Reporting by Neil Robinson; Editing by Peter Rutherford and Pritha Sarkar)
