After aggravating his original injury and being withdrawn at half-time against Braga in late October, the Japan star's "progress has been slow" and will not face Manchester City
The development represents a worsening of the initial prognosis, which had originally projected the Japan international to recover from his knee injury a month after aggravating it in late October.
Kagawa had made a bright start to his career at Old Trafford following his summer switch from Borussia Dortmund, scoring two goals in his first six games.
But Sir Alex told reporters ahead of the Red Devils' Champions League tie with Galatasaray: "His progress has been slow."
Among other games, the attacking midfielder is now expected to miss the derby with Manchester City at the Etihad Stadium on December 8.
The Scot confirmed more positive recovery news, however, when revealing that Phil Jones will start against the Turkish side on Tuesday night.
"He's done enough training to make sure his fitness is OK," the veteran manager declared. "I'm looking forward to bringing him back."
Eighteen-year-old Nick Powell will also feature, with Sir Alex adding: "He's played in front of 76,000 as a substitute but this will be his first European away game. It'll be a big day for him."