The defender could miss trips to Spurs and Man City, but may return in time for the home game against Queens Park Rangers.
The Toffees centre-back was stretchered off in the latter stages of their 2-1 defeat to Manchester United at Old Trafford, which meant he missed England duty, and is due to see a specialist to assess the true extent of the ankle injury suffered by the 20-year-old as he landed awkwardly after contesting a header in the latter stages of the contest.
It is a big blow for Everton, and Stones, with the young defender impressing at the heart of their back line in recent weeks, and his stock was growing considerably.
If out for two months, there are a couple of big games the defender could miss a trip to Tottenham Hotspur at the end of November - as well as three Europa League games - Lille home and away, and the trip to Wolfsburg. He may also miss the home game against Krasnodar which concludes the side's participation in the group stages.
The chances are that Stones could also miss the trip to Manchester City on December 6, whilst a return for the home game against Queens Park Rangers on December 15 may be a realistic target.
Roberto Martinez's side are a little short on defensive numbers at the moment. Seamus Coleman has missed recent games because of a head injury whilst Sylvain Distin has been nursing a thigh injury. Given that Stones can cover both the right-back and centre-back positions, he will be keen to have the pair back in the squad immediately after the international break.