It came at a time in January 2017, when Liverpool were second in the table and had just dropped points in a 2-2 draw away at Sunderland to fall further behind eventual Premier League champions Chelsea.
Klavan claimed that under growing pressure, Klopp was prepared to fight with Milner there and then in the dressing room to prove his point.
Klavan said on the Betsafe Eesti podcast: "The biggest pressure was around Christmas and at the beginning of January. One time, I don't remember who we played against. Maybe it was against Sunderland away.
"Klopp and James Milner almost started a physical fight. It was the period when there was Boxing Day in England and they almost started fighting but eventually Milner stepped back.
"We saw from his eyes that Klopp would have gone all the way just to prove his point. It was the most stressful time for him as well. He was under a lot of pressure.
"It was my first and Klopp's second year. What are you going to do? You're supposed to be a good coach. You are Jurgen Klopp who will make superstars of all players. The pressure started getting to him.
"It's not supposed to be easy, but if you don't deliver against a team like Sunderland, then you can forget about the places you had hoped for."
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