The Croatian has been hampered by the injury for a while and sat out last month's 1-0 FA Cup final loss to Crystal Palace.
"Mateo will spend the summer rehabilitating from the operation, and everyone at the club wishes him a speedy recovery," the club said in a statement.
The 31-year-old was not included in the squad for Croatia's World Cup qualifiers against Gibraltar on June 6th and the Czech Republic on June 9th.
Kovacic scored seven goals in 42 appearances for City this past season across all competitions.
City kick off their Club World Cup campaign in the United States on June 18th against Wydad AC.
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