Khuzwayo sustained the injury in his side's 3-0 win over Stellenbosch FC in the Nedbank Cup a fortnight ago.
“Brilliant will undergo surgery on Friday,” team physiotherapist Dave Milner told the club's website.
Khuzwayo will likely miss the team's crunch clash with champions Mamelodi Sundowns at FNB Stadium on April 1.
First-choice keeper Itumeleng Khune is currently with the national squad, and will captain Bafana Bafana in the friendlies with Guinea-Bissau and Angola on Saturday and Tuesday.
Amakhosi also have inexperienced keepers Brylon Lee Petersen and Luthando Sixhaso on their books.