The Parkhead club confirmed the 19-year-old left-back suffered the injury in training.
His place for the Ladbrokes Premiership clash with Aberdeen at Pittodrie on Saturday lunchtime will likely be taken by experienced Honduran Emilio Izaguirre.
Former Celtic youth player Tierney will also miss Scotland’s crunch 2018 World Cup qualifier against England at Wembley on November 11 and the Betfred Cup final against the Dons at Hampden Park on November 27.
Tierney will be absent for the rest of Celtic’s Champions League Group C fixtures against Borussia Monchengladbach, Barcelona and Manchester City.
Scotland boss Gordon Strachan is already without Hull left-back Andrew Robertson, who has been ruled out of the trip to London with a calf injury, which could mean a return of Rangers captain Lee Wallace who was in the squad for the 3-0 defeat in Slovakia earlier in the month.
Aberdeen defender Graeme Shinnie, Stephen Kingsley of Swansea City and former Dundee United full back Barry Douglas, now with Konyaspor in Turkey, could also come into contention for Strachan’s squad which is expected to be named on Wednesday.
A statement on the Celtic website read: “Celtic have suffered an injury blow with the news that Kieran Tierney is out of action for two months after he suffered ankle ligament damage in training.
“The 19-year-old has been in superb form this season for the Hoops and has been an ever-present in Brendan Rodgers’ starting XI before missing out on the midweek win over Ross County.
“Everyone wishes Kieran a speedy recovery from injury.”
On his official Twitter account, Tierney wrote: “Thanks a lot for all your messages and support! Means a lot to me as always. God bless”.