The 18-year-old was substituted at half-time of Saturday's 5-2 LaLiga win over Real Betis at Camp Nou.
Spain international Fati was sent for tests, which have shown a tear in the meniscus of his left knee.
The club confirmed treatment will begin over the coming days.
LATEST NEWS ‼ Tests carried out this afternoon have shown that Ansu Fati has an internal meniscus tear in his left knee. The treatment to be followed will be determined in the coming days. pic.twitter.com/CF3FZaur5P— FC Barcelona (@FCBarcelona) November 7, 2020