The 25-year-old, who has 27 caps, was taken off on a stretcher during the second half and has since had surgery.
The Cherry and Whites say Morgan will "commence an appropriate rehabilitation programme over the coming weeks."
England's campaign starts against Wales on 6 February and finishes on 21 March.
Coach Stuart Lancaster is set to announce his squad for the Six Nations after next weekend's round of European club games, and Morgan's injury could open the way for Saracens' Billy Vunipola to regain his starting place.
Alternatively, Wasps' in-form captain James Haskell could fill in at number eight.
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Morgan was one of England's most impressive performers during the autumn internationals, scoring a try after coming on as a replacement against South Africa, before being promoted to the starting XV against Samoa and scoring two tries in the win over Australia the following weekend.
He signed a new contract with his hometown club last week, having joined them in 2012 after a spell with Welsh regional side Scarlets.
Morgan made his England debut against Scotland in February 2012, turning down the chance to represent Wales, for whom he would have qualified on residency ground from his time at Parc y Scarlets.
Gloucester and England number eight Ben Morgan is likely to miss the whole of the Six Nations tournament after fracturing his left leg in his club's 24-23 win over Saracens on Friday.