The 30-year-old had already been a doubt for the opening games due to issues with the same area, but he has since hurt his other ankle in training.
Farrell's club Saracens have revealed that he will miss at least two months, with England head coach Eddie Jones acknowledging that the development is "a blow" to the squad.
Speaking on the injury, Jones said: "Owen is going to have an operation so he'll be out of the Six Nations. I know a surgeon saw him on Sunday. They're making the plan and hope to operate as soon as possible. I'm not a doctor, but I would imagine the rehab is going to be similar to his previous injury.
"It's a massive blow for him personally and for the team it's a blow, but it's obviously an opportunity for other guys to step up into leadership roles and for other players to play for that 12 jumper."
England begin their campaign with a showdown against Scotland on February 5. body check tags ::