Reihana, who injured his right knee in a tackle on Olly Morgan just six minutes into Saturday's game, will undergo surgery on Thursday.
Saints medical director Cliff Eaton said: "The prognosis for Bruce's injury was better than we originally thought but it will need surgical intervention and we are fortunate enough to have the skills of orthopaedic surgeon Bill Ribbans at our disposal."
It is another significant injury blow for Northampton, who discovered on Monday that Steve Thompson faces six weeks on the sidelines with a torn calf muscle.
Centre Jon Clarke is also out until December after breaking his ankle on the opening day of the new season.
Director of rugby Budge Pountney will have to name a new skipper for the Saints' EDF Energy Cup game against Worcester at Franklin's Gardens.
Northampton also have injury concerns over hooker Matias Cortese, who sprained an ankle in last night's Guinness A League defeat to Leicester.
Lock Tom Powell faces a scan to determine the extent of the damage to ligaments in his ankle.