The 35-year-old right-back suffered the injury playing in Tuesday’s French Cup final against Les Herbiers, coming off four minutes from the end as PSG beat the minnows 2-0.
The French champions said in a statement that tests on Wednesday had shown damage to the anterior cruciate ligament in his right knee.
They said he will require “at least three weeks of treatment before reassessing the need for surgery.”
The injury is a potentially massive blow to Alves and Brazil, with national coach Tite set to announce his squad for the World Cup next Monday.
Brazil have already been sweating on the fitness of Alves’ PSG team-mate Neymar, who underwent surgery on a foot injury two months ago and has not played since.
Tite must now decide whether he can afford to take a risk on the fitness of the former Barcelona star Alves, who won the 38th major honour of his career on Tuesday.
There may be time for him to recover, though, as Brazil play their first World Cup match against Switzerland in Rostov-on-Don on June 17, five and a half weeks away.
They will also take on Costa Rica and Serbia in Group E in Russia.
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