The Reds beat Tottenham 2-1 at Anfield on Wednesday night to reclaim top spot in the Premier League thanks to late header from Roberto Firmino.
But after the game Klopp appeared to be riled by a comment from Mourinho after the final whistle.
The top bosses clashed after they shook hands following the game, with Mourinho appearing to say something to Klopp which caused the Liverpool chief to double-take.
Mourinho later revealed to the press he had told Klopp 'the best side lost', which had caused the Reds manager to smile in confused amusement before confronting the Portuguese.
In his post-match press conference, Mourinho then took aim at Klopp again for his his pitchside behaviour, saying when asked about Firmino's winner: "I don't want to speak about it, I prefer not to speak. If I speak about that I've to speak about the behaviour of some coaches on the touchline."
Klopp has been criticised for his often sulky figure on the touchline this season, with the boss particularly hammered for his comments after Liverpool's draw with Fulham, when he took aim at the Cottagers for trying to stop the champions from playing their game.
And former Liverpool defender Lawrenson admits the German doesn't look at all happy.
Speaking on talkSPORT 2, host Paul Coyte asked the Liverpool legend: "We saw a very happy Jurgen Klopp last season, for obvious reasons because they weren't losing games.
"But then we see the other side, when there's a little pressure on he does seem to get a little bit touchy and a bit upset, doesn't he?"
"Oh crikey, yeah," said Lawro.
"He absolutely totally lives it, doesn't he.
"I think at the moment, although they won last night and he'll be delighted with the way they won, he doesn't look happy with football.
"I'm not saying he doesn't look happy with Liverpool, the team or the results, but he just doesn't look happy with football.
"There's VAR, and everybody has peeved about VAR, the lack of fans, and the big thing is the five sub rule.
"I think he's got a point and everyone will come round to that. When he first said about having five subs I thought no, it just makes the stronger teams even stronger.
"But when you look at football now and the amount of games we're having, they've had the whole European group campaign in about four of five weeks, which they've had to do, as well as league and cup games - it's mad.
"So I think this could be a special year and I do think they should bring it in.
"And the things with fans, without spectators the game is just completely different. Even last night with 2,000 at Anfield it's just very, very strange.
"Klopp, Mourinho, They want a perfect world and they want to win every week, all managers are the same."
Liverpool legend Mark Lawrenson has told talkSPORT Jurgen Klopp seems 'unhappy with football' after the manager's latest touchline clash with Jose Mourinho.