Son Heung-min has scored 96 goals in 245 Premier League appearances for Tottenham
Son, 30, suffered the injury during Tottenham's 2-1 win in Marseille in the Champions League on Tuesday.
"Following surgery, Son will commence rehabilitation with our medical staff and we shall update supporters further in due course," said a Spurs statement.
South Korea's first game in the World Cup is against Uruguay on 24 November.
Tottenham have not said when Son will have surgery and have not put a timescale on how long the former Bayer Leverkusen forward will be sidelined for.
Manchester City playmaker Kevin de Bruyne broke his nose and eye socket in the 2021 Champions League final but played for Belgium in the delayed 2020 European Championship 19 days later.
Son has scored 35 goals in 104 appearances for his country, who are in Group H in Qatar alongside Portugal, Ghana and Uruguay.
In the more immediate future, the injury is a blow for Spurs, who host Liverpool in the Premier League on Sunday (16:30 GMT).
Antonio Conte's side are third in the table - five points behind leaders Arsenal having played one game more - then face Nottingham Forest away in the third round of the Carabao Cup on 9 November and Leeds United at home in the league on 12 November before the break for the World Cup.
Son has had his form questioned this season but scored a hat-trick after coming on as a substitute in the 6-2 win over Leicester City on 17 September and netted twice against Eintracht Frankfurt in the Champions League on 12 October.
-- Courtesy of BBC Sport