Murray limped off late in the first half of what was his 100th Test for his country, Ireland's gritty 19-16 Test victory over world champions South Africa on Saturday.
Munster star Murray will miss next Saturday's game against Fiji and Ireland's final Test of the year a week later when they host Australia.
Murray looked distraught when he came off as he had been handed an opportunity by head coach Andy Farrell to reclaim the first-choice role he held for over a decade before being succeeded by Jamison Gibson-Park.
Gibson-Park impressed when he replaced Murray on Saturday despite it being his first competitive rugby since July.
Farrell will choose between Gibson-Park and Murray's Munster rival Craig Casey for the starting spot against Fiji.
The two other players who came off injured, Stuart McCloskey and Tadhg Furlong, have both been given a clean enough bill of health to train with the squad this week.
Meanwhile, captain Johnny Sexton is "recovering well" from a dead leg