Farrell injured his ankle in the win over Australia.
Owen Farrell, the England captain, is set to miss the Six Nations opener against Scotland due to injury.
Farrell, who hasn't played since November because of an ankle problem, was retained as England's captain by Eddie Jones.
He was hoping to return against London Irish in the Challenge Cup on Sunday, but has now been ruled out.
England will play Scotland on February 5.
Farrell will see a specialist early next week and is not expected to join up with England in their training camp.
England will announce a squad update on Monday with Lawes favourite to take over the captaincy in Farrell's absence.
'I don't know where he fits' - Matt Dawson on Owen Farrell.
Owen Farrell was ruled out of the game against London Irish due to an injury.
Owen was back in full training after recovering from an ankle injury he sustained in the Autumn internationals.
He will see a specialist early next week.
Marcus Smith will take over as England's fly-half after impressing during the November Tests, with Farrell's involvement for the start of the tournament in doubt.
George Ford could be recalled despite being surplus to requirements through the autumn, while Joe Marchant is one of the centre options available to Jones.
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