In the middle of July Klopp's side defeated the Blues 5-3 before lifting the Premier League title, but during the match a video was taken which showed the two managers, and a couple other staff members, exchanging heated words between the two dugouts.
The video went viral afterwards, and due to the game being played without spectators every word and profanity could be heard.
Both managers said they were simply defending their team and staff, but Lampard has now said he regrets what happened.
"Yeah, I did regret that," the former Chelsea midfielder said on the High Performance podcast.
"And when that broke the next day – I clearly felt it as I was doing it – but when it broke the next day and a friend of mine sent it to me in the morning, I was a bit embarrassed by it.
"Because I was in the moment, but in the moment, I felt we turned up and it was the easiest day for Liverpool ever.
"They won the league, they went goals up early in the game. And a few things happened with the bench, I’m not going to go into detail, but my feeling was, 'I want to protect my club'"
Lampard says he respects Klopp and he will aim to put the situation right when they two meet again this campaign.
"And I didn’t have a problem with Liverpool celebrating," he continued.
"Luckily enough, I’ve been there. I’ve been with Chelsea where we won the league quite early one year, and you can sit there and everything feels great.
"My feeling was, I want to be there where they are. I definitely didn’t mean any disrespect to Jurgen Klopp, because I’ve got huge admiration for them.
"But what I felt had gone on, it was an impulse reaction, which I will happily – when I do see him, I’ll put that one right."