The scrum-half damaged his shoulder in a second team match for Northampton on Monday.
It was only Dawson's second start since the World Cup final on 22 November following a persistent calf injury.
The 31-year-old played against Leicester in the Premiership before Christmas, but lasted just 28 minutes.
The news will troble England boss Clive Woodward. Dawson is one of nine members of the World Cup-winning squad to be nursing injuries, with Mike Tindall, Iain Balshaw, Mike Catt, Jonny Wilkinson, Richard Hill, Lewis Moody, Julian White and Martin Corry all struggling.
And Sale's Charlie Hodgson - the likely back-up to Wilkinson at fly-half - is awaiting the results of a scan after injuring his knee. England's Six Nations campaign opens against Italy in Rome on 15 February.
The Saints star scrum half will miss the make-or-break Round 6 tie against the Sacrlets at Franklin's Gardens.