The 24-year-old initially sustained the injury on his debut in the opening game of the season against Liverpool.
But after playing through a hectic spell of 10 games plus an international break with Germany, Koch suffered further damage in the 3-1 loss to the Blues at Stamford Bridge on Saturday.
"Unfortunately, following a collision in the Chelsea game, he suffered further damage to the knee and the club have made the decision that surgery is required to remedy the problem," Leeds said in a statement.
It is another blow for Leeds boss Marcelo Bielsa, who has only two fit senior centre-backs in his squad.
Diego Llorente and Liam Cooper are Bielsa's current remaining options, with no indication yet as to how long Koch will be sidelined for.