Goode impressed for England in the autumn internationals and looked to have cemented his place ahead of Harlequins' Mike Brown.
But Sarries boss Mark McCall admits it's touch and go whether or not the full-back will be fit in time for England's first game against Scotland on February 2.
"Alex has a shoulder injury - a tear in his shoulder - and at the minute we are looking at a non-surgical route," McCall explained.
"There's a chance he might be fit in two or three weeks time but it all depends on how his rehab goes in the next couple of weeks.
"If he makes the progress we hope he makes he's got a great chance of being available for the Six Nations.
"I think England may want him to play in a game or two, maybe in the LV=Cup before that, but he's got a chance - it all depends on how these next couple of weeks go."
McCall made it clear England have been aware of the situation throughout and were consulted over the decision.
He added: "England have been in the loop the whole way - there's no issues there at all - and it was felt that the non-surgical route was the best route to attempt."