The 23-year-old Croatian was substituted early in the second half of Sunday’s 1-1 home draw against Chelsea.
"The defender will have surgery later this week, and assessment remains ongoing to ascertain the full extent of the injury and expected prognosis," City said in a statement.
City are second in the table on 42 points, six behind leaders Arsenal. Their defender Ruben Dias also went off due to injury in the Chelsea match, and England centre-back John Stones is sidelined too.