Klopp was so furious with a tackle by Sunderland’s Jeremain Lens on Liverpool’s Mamadou Sakho that police stepped in to calm the tension between the two benches.
Allardyce also had to apologise after claiming Liverpool’s injury problems were down to Klopp’s demanding style of play.
The pair do not have long to wait for their next touchline encounter, with the first Merseyside derby of the season taking place at Anfield on December 10.
But Klopp would be happy to see Allardyce succeed because a strong Everton team would be good for the area.
“With Sam I have kind of a history, when he was at Sunderland and he said something after the game then he had to row back, not for me, only because we are in the public,” Klopp said on Friday.
“Then he was the manager of the England national team and we had a very good relationship in his short term.
“I’m happy for him, it’s good for us when Everton is in a good moment – they shouldn’t be in a better moment than we are, but in a good moment it’s important for us and I think the whole city.”