Nakamba picked up the injury in last weekend’s 1-0 defeat at Liverpool and subsequently sat out 2-0 victory at Norwich City in midweek.
The 27-year-old Zimbabwe international has been in fine form under a Villa side that has been resurgent under new manager Steven Gerrard.
Nakamba, who joined Villa from Club Brugge in the summer of 2019, has made 14 appearances across all competitions this season.
“He’s had successful surgery,” said Aston Villa manager Gerrard.
“The surgeon is really happy with his operation and how it’s gone and now he’ll start his recovery process and recovery plan.
“He’s going to be missing for somewhere in the region of 12 weeks if it is a fantastic rehab or it could be up to 16 weeks.
“It will be somewhere in the middle of 12-16 weeks.”
Nakamba will also miss the upcoming Africa Cup of Nations as a result and will be a major absentee for Zimbabwe.
The Harambee Stars are in a group that features the likes of Senegal, Guinea and Malawi.
