Alves originally picked up the injury during PSG's 2-0 win over Les Herbiers in the Coupe de France final on Tuesday.
The Brazilian national team confirmed that he will be unavailable for the tournament on its Instagram account on Friday, dealing one of the favorites a massive blow just over a month before the competition begins in Russia.
Alves played for Brazil in the last two World Cups and has 107 appearances for the team since his 2006 debut.
PSG issued a statement of their own, adding that club and Brazil medical staff have agreed surgery is the best option for Alves.
Brazil play warm-up matches against Croatia on June 3 and Austria on June 10, before kicking off its World Cup campaign on June 17 against Switzerland. "Footballers hate missing games but I know first-hand that missing World Cup games is the worst feeling", Pele, who was forced out of the 1962 and 1966 World Cup tournaments because of injury, said on twitter.
Meanwhile, reports coming from the south American country claim the defender will not be able to travel to Russia at all.
Gaspar also said Brazil coach Tite has enough information to pick an adequate substitute at right back.
"The technical team of the Brazilian national team feels for the event and wishes Daniel Alves a prompt recovery so that his technique, commitment and leadership are back at the service of the national team as soon as possible".
Former Brazil player Junior believes Danilo is the most likely replacement.
Tite is due to name his squad on Monday.
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