Alves, 35, limped off near the end of PSG's 2-0 win over third-tier Les Herbiers with what was diagnosed as an anterior cruciate ligament (ACL) injury.
Although his spokesman said he was optimistic of making the tournament, which kicks off in Russia on June 14, fears he may miss out grew when his club said it will be the end of the month before they know for sure how serious the injury is.
"After suffering a knock on his right knee during last night's @coupedefrance final, @DaniAlvesD2 underwent exams that confirmed an ACL injury," PSG said on Twitter.
"The player will be evaluated over the next three weeks before determining whether surgery is necessary."
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Brazil coach Tite is scheduled to name his 23-man squad for the finals in Russia on Monday.
Brazil play warm-up matches against Croatia on June 3 and Austria seven days later before heading to Russia where they will face group rivals Switzerland, Serbia and Costa Rica.
(Reporting by Andrew Downie; Editing by Ken Ferris and Ed Osmond)
