The former Juventus and Italy boss says that goes with the territory of working at Chelsea and does not happen at other clubs. "But remember, when you wind Mourinho up, he's very clever in these debate situations".
"I think when a person insults another, you can expect a response or you can expect contempt, silence", Mourinho said.
Conte has had some issues with the board about the Chelsea's transfer budget- vehemently arguing that the funds supplied are not enough to win the league.
Mourinho tried to tread a diplomatic path around the subject at his press conference for Monday's clash with Stoke, but said there are "some players in the football word" you have to try to sign if clubs have the chance. "I don't think so", said Conte.
"There is something odd that after the first game we lost against Burnley the press pushed quickly to sack me".
Asked if the League Managers' Association (LMA) should now step in to try to fix the damage between the pair, Conte said: "This is not a problem from the club, it is a problem between me and him. I continue to work and be professional and have good result".
Chelsea were held to a goalless draw by Arsenal at Stamford Bridge in the Carabao Cup Semi-final and will get a replay at the Emirates. He's a great player and he knows very well in my first season in Juventus we tried to buy him from Udinese. "Then the club has to decide which solution the club prefer". I don't ask to change the situation.
"He played for the national team and is scoring with regularity in this league and is a good player, but this different".
Both Mourinho's and Conte's contracts with their respective clubs expire at the end of the 2018/19 season, with the duo insisting they intend to see out their current deals before moving on to another club.
Capello told Sky Italia when asked about the feud: "Conte didn't understand who he was dealing with".
Meanwhile, former Manchester City manager Roberto Mancini believes the feuding pair will soon make up. This awoke something in Antonio Conte, and the club wasted no time in looking for another forward to add to their squad.