Tottenham won 5-4 on penalties in the Carabao Cup last 16 after Erik Lamela's equaliser cancelled out Timo Werner's first-half opener in normal time.
During the London derby at Tottenham Hotspur Stadium, Chelsea boss Lampard and Spurs counterpart Mourinho were involved in a heated exchange in the first half.
Mourinho and Lampard enjoyed incredible success together as manager and player at Chelsea, where the pair won two Premier League titles together among other honours.
But there was no love lost between the pair during the Cup tie, though both were quick to praise each other after the match when asked about the incident.
"We were having some fun on the line," Lampard told reporters after Mason Mount's missed penalty. "I get on well with Jose and I think he said that to me.
"I commented on the fact that he seemed to speak more with the referee than he speaks with his players so it is just a moment with us.
"I have a huge respect for Jose, no matter what he says on the touchline. I think he was feeling the sense of the first half and how it was. All was well at the end of the game and that will never change."
Mourinho added: "With Frank the most important thing is my feelings towards him rather than any words we exchanged. I owe him everything he gave me, he always gave me everything he had as a player and I never forget that.
"My feelings towards him will always be feelings of how much I owe him for how much of an incredible player, friend he was.
"The only thing I was telling him, just an opinion of an old coach to a young talented coach, which was when the players need us it is when they are losing, when they are winning we don't need to be the protagonists of the touchline, we need to be there when they are losing.
"In the last match when they were losing 3-0 [against West Brom] I felt really sorry for him because he was sad and quiet in his chair. In terms of football I can't teach him anything, he knows football, it is just an opinion, stay on the touchline when your team is losing and stay calm when your team is winning."
