Tests have confirmed the Brazilian suffered an ankle sprain in Thursday's win over Getafe but could be in contention to face Manchester City in the Champions League next month.
The Brazilian was helped off the pitch during Thursday's 2-0 Copa del Rey win over Getafe after landing awkwardly on his right ankle during the first half.
Head coach Gerardo Martino revealed after the match the damage was not as serious as first feared, and further medical checks have since confirmed he will be sidelined for potentially less than a month, meaning he could be in contention to face Manchester City in the Champions League last 16.
"Tests performed on Friday confirmed that the player Neymar Jr. has a sprain in the peroneal tendon of his right ankle," a statement on the club's official website confirms. "The estimated recovery time is three to four weeks.
"Neymar will certainly miss the next three Liga games against Levante, Malaga and Valencia, and the quarterfinal of the Copa against Levante."
The news means Neymar is likely to be in contention to face Rayo Vallecano on Feb. 18 before the trip to the Etihad Stadium for the Champions League last 16 first leg with City two days later.
Barcelona triumphed 2-0 on Thursday thanks to a double from Lionel Messi, ensuring the club progressed to the last eight with a 6-0 aggregate win.