The 30-year-old Harlequins player aggravated an existing shoulder problem at Worcester last Sunday and has been ruled out of Saturday's Premiership game against Sale.
"I'm seeing a specialist on Monday and will know more after that," he said.
"It's all up in the air at the moment and it's hard to know where I stand."
Eddie Jones' side have a number of injury worries going into the tournament, where their first game is against France on 4 February.
"Chris will see a specialist early next week to determine the best course of action for his recovery; either an extended period of rest and rehabilitation, or an operation," said the club in a statement.
Billy and Mako Vunipola have been ruled out with knee injuries, while Joe Launchbury has a calf problem and will be unavailable for Les Bleus' visit to Twickenham.
James Haskell and George Kruis are doubts with a foot injury and fractured cheekbone respectively.
And current skipper Dylan Hartley will need to prove his fitness after serving a six-week suspension.
England flanker and former captain Chris Robshaw could miss the Six Nations because of a shoulder injury and is to see a specialist next week.