The Australia international, 24, last played for the Scottish Premiership leaders on 17 December against Dundee.
He came off at half-time of that match, which Celtic won 2-1, and aggravated the injury in training as he sought to get back to full fitness.
"His foot stuck in the ground and he did a full twist on it," said Celtic manager Brendan Rodgers.
"It was unfortunate, it was right at the end of training, the same spot where he had it before.
"He had to have an operation. [He] will be out probably for the next few months.
"It was either going through the same procedure as before or have an operation so he went to London for an operation and it is all done and he is on the way to recovery now.
"It is a big disappointment because he is a player I was looking forward to seeing back.
"He will be back for the end of the season, it is a disappointment that he will miss up to eight weeks."
Rogic, who joined Celtic in 2013, has scored 10 goals in 27 appearances this season.
Midfielder Stuart Armstrong and forward Leigh Griffiths are nearing returns from injury but will not feature against Inverness Caledonian Thistle in Saturday's Scottish Cup fifth-round tie in Glasgow.
Meanwhile, Rodgers says the club are nearing agreement over a new contract for goalkeeper Craig Gordon.
Celtic fended off interest in the 34-year-old from Chelsea during the January transfer window.
"I have spoken to the club and it is very close, just wrapping up one or two details I believe, 99.9% is all agreed," added Rodgers.
"It will be great news once that is complete. That is the main signing in the January window as such.
"He is a real pivotal part of how I want to work. I have made it clear since I came in how I wanted to impose my style and the importance of the goalkeeper within that.
"And I look at him now, seven months on from when I came in, and he is a totally different character both in personality, character and his output in the team.
"So to have that assurance for the next few years is very important for the club and another sign of players who are committing to us because they believe in what we are doing."
Celtic midfielder Tom Rogic faces a "few months" more on the sidelines following an ankle operation.