The Liverpool manager revealed he had extended chats with the Blues boss before and after Tuesday’s 1-1 draw at Anfield.
Despite their rivalry in the Premier League and on the sidelines, the passionate pair were able to talk harmoniously.
However, Klopp’s relationship with Manchester United’s Jose Mourinho does not appear to be on as good terms.
The Portuguese coach hit out at how Klopp escaped punishment for shouting at four official Neil Swarbrick in the Liverpool and Chelsea clash, but the German refused to engage in a war of words.
Jurgen Klopp and Antonio Conte spoke for 10 minutes either side of Liverpool and Chelsea's match
I talked 10 minutes before the game to Antonio Conte and 10 minutes after
“Honestly, I have no time to think about things like this,” Jurgen Klopp said. “Why are we talking about other managers?
“If you want a positive story, I talked 10 minutes before the game to Antonio Conte and 10 minutes after the game to him and it was really interesting.
“We don’t know each other but we have something to talk about.”
The importance of Chelsea and Liverpool's game this week did not appear to come between the managers
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Pep Guardiola’s arrival at Manchester City in the summer also reignited a rivalry with Klopp from their time at Bayern Munich and Borussia Dortmund respectively.
But even with their Bundesliga history, the Liverpool boss still has conversations now when they meet in English football, whereas Klopp admits his relationship with Mourinho may never reach that level.
Klopp added: “Pep Guardiola? We are not friends but we know each other, so we talk when we meet.
Jurgen Klopp and Jose Mourinho clashed when Liverpool and Manchester United drew 1-1 at Old Trafford
“But you cannot have relationships like this with all managers. And if I don’t have one, there is not time to start having one.”
Guardiola’s City trail Klopp’s Liverpool only on goal difference in the Premier League, and they meet again at the Etihad on March 19.
Meanwhile, the Reds have already faced sixth-placed United and table-toppers Chelsea twice this season.