The German took to the field to embrace goalkeeper Alisson Becker after substitute Divock Origi scored a stoppage-time winner at Anfield.
That reaction led to the FA issuing him with a misconduct charge and giving him until Thursday to respond, but he is thought to have graciously accepted the charge.
Klopp is expected to be handed an £8,000 fine and a warning over his future conduct by English football's governing body.
However, there is good news for Liverpool in that Klopp will not be hit by a touchline ban.
The former Borussia Dortmund boss immediately apologised for his actions following the game, saying: "I have to apologise because I didn't want to be disrespectful - but I couldn't stop myself.
"If I could describe how I felt when the goal went in, then I would have control over it. We don't want to look for excuses but that's how it is.
"The plan was not to run. When I chose to stop, I was close to Alisson. He was surprised."