Those words prompted an acerbic response from Mourinho referencing Conte's history of facing accusations relating to match-fixing.
Conte's jab followed Mourinho's assertion that other managers act like "a clown" on the touchline, per Samuel Luckhurst of the Manchester Evening News.
It's prompted Conte to call on Mourinho to remember his own history: "I think that he has to see himself in the past, maybe he was speaking about himself in the past, yeah?"
Conte continued: "And sometimes I think there is, I don't know the name—'demenza senile'?—when you are a bit [Conte taps his temple], when you forget what you do in the past."
Mourinho pulled no punches with his retort following United's 2-0 win over Derby County in the 2018 FA Cup third round at Old Trafford on Friday night.
The 54-year-old blamed reporters for asking a leading question to Conte and demanded an apology, per BBC Sport's Simon Stone. He also had this to say about Conte's past: "I don't need Chelsea manager to say I made mistakes in past and will in future. What I will say is I will never be banned for match fixing."
Conte was acquitted in 2016 of charges of not stopping a match-fixing scandal when in charge of Sienna in 2011, per Reuters (h/t Sky Sports).
Conte's reference to Mourinho's past antics on the touchline is a telling one. Mourinho has a history of extravagant celebrations, dating back to when his Porto side knocked United out of the UEFA Champions League in 2004.
It was an achievement made into an iconic image of Mourinho sprinting down the touchline at Old Trafford to celebrate with his players.
Yet Mourinho's most infamous act of celebratory bravado came in 2010 when Inter Milan eliminated Barcelona from Europe's premier club competition. Mourinho rejoiced by charging onto the Nou Camp pitch to cheer, which incited anger among Barca's players, particularly goalkeeper Victor Valdes.
ESPN FC correspondent Sid Lowe recently reported (h/t Lewis Winter of the Daily Express) how Mourinho stored a "life-size, cardboard cutout version of that image" in his office while in charge at Real Madrid.
It is a bit rich for a manager with Mourinho's history of bombastic celebrations to be critical of another, even if Conte has become famous for his passionate reactions during matches.
Conte believes Mourinho, who won three Premier League title in two spells at Chelsea, is obsessed with his former club, per Kajumba: "There is a person who continues to look here. He went away but continues to look here."
The rivalry between the two is unlikely to abate anytime soon with both clubs vying for second place behind runaway leaders Manchester City.