The Sarries fly-half is to have surgery to treat an injury to the base of his thumb and his recovery could take up to 10 days.
With England beginning their tournament in Dublin on February 2, that would leave Farrell little time to recover.
Speaking to BT Sport, Saracens director of rugby Mark McCall said: “It’s a simple procedure and he should be back in seven to 10 days.”
Earlier this week, Farrell was named sole captain after co-skipper Dylan Hartley was ruled out of the start of the tournament with a knee injury.
Team news | Owen Farrell (wrist) is out of today’s match with Glasgow.
— Saracens Rugby Club (@Saracens) January 19, 2019
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