The unsavoury remarks were kicked off when Mourinho implied he did not need to celebrate in the fashion that Liverpool's Jurgen Klopp or Conte do when their team scores a goal or wins a game in order to illustrate his passion.
Conte - who was acquitted of sporting fraud charges in 2016 over accusations he failed to report episodes of match-fixing while in charge at Serie B side Siena in 2011 - said he had no regrets about his response to Mourinho's attack.
He said: "I think it's not important, the association, you understand. We both said these things and we'll see what happens in the future".
Asked if the League Managers' Association (LMA) should now step in to try to fix the damage between Conte and Mourinho, the Chelsea boss said: "I think it's not important, the association, you understand".More news: Trump Appears To Forget The Words To The National Anthem
"He said a serious words, using serious words". I won't forget this.
"It's not a problem for the club, it's a problem between me and him". "Now I stop. Stop".
Conte's stance ensures that all eyes will be on the Old Trafford touchline when Chelsea take on United in the Premier League on February 25, but the Italian has a more pressing challenge to navigate first as he prepares for a two-legged Carabao Cup semifinal tie against Arsenal.
Ross Barkley is unlikely to be involved however, though Conte hailed the new recruit for settling quickly to his task following his transfer from Everton.
"I think it's not important".
"I don't think so".