The former Great Britain rugby league captain, who was recalled to the side after being dropped following the record 36-0 defeat by South Africa last month, underwent an MRI scan on Thursday.
England head coach Brian Ashton will not announce Farrell's replacement until Friday.
Ashton, though, would appear to face a straight choice between promoting Olly Barkley off the bench or recalling 36-year-old Mike Catt for his first appearance since the South Africa drubbing.
Barkley had a poor game defensively against Tonga six days ago, while Catt offers considerable experience and is a survivor of England's successful 2003 World Cup campaign.
The loss of Farrell is a cruel blow both to the player and his team, with Ashton having selected him for the "direct approach" he would have brought.
Farrell's injury denies him a chance to play in the biggest game of his union career since switching codes and joining Saracens.
His progress in the union ranks has been repeatedly stalled by injury problems, seriously hampering his progress with club and country.
The 32-year-old made his England debut against Scotland during last season's RBS 6 Nations, while he produced a try-scoring display during Friday's victory over Tonga.
England's World Cup campaign has been dogged by injuries, with the likes of Barkley, Jonny Wilkinson, Jason Robinson and Mark Cueto among those who suffered setbacks at various stages.
Barkley would offer England a considerable kicking option alongside Wilkinson in the number 12 jersey against Australia, yet Catt might be more suited to Ashton's plan.