The 36-year-old centre back was substituted at half-time in the 2-0 home win over Manchester United on Sunday and manager Roberto Martinez has confirmed that the player will not be available for the trip south.
"It's a little bit too early to know how long he is going to be out," said the Spaniard, who already has injured Belgian forward Kevin Mirallas ruled out for the rest of the season.
"It's a soft tissue injury with his hamstring and I do feel that in the next 48 hours we will have a clearer idea of whether he is going to be available before the season ends."
Everton are fifth in the standings, a point behind Arsenal in the battle for a place in Europe's top competition next season, and chasing their fourth successive away win.
Martinez said it was also unlikely that on-loan striker Lacina Traore, who has been out since February, would play again this season with three games remaining.
"I don't think we'll see Lacina again this season. It would be a surprise if he's fully fit before the Hull City game," he said of the Ivory Coast international, who featured in only one game before injuring his hamstring.
"He's recovering well and clearly he'll be fully fit for the World Cup if he's considered for selection. But at the moment I think any games in this league campaign are going to be a little too early for him."
The Toffees have Phil Jagielka back in training after being sidelined since February with a hamstring injury but Martinez said the defender would not be rushed into action.
"We want Phil back as quick as we can but not because of Sylvain," the Spaniard said.
"Phil is going to be fully fit very soon and I do expect that he will probably be able to make the squad against Manchester City on May 3 - that's always been the plan."
- Sports & Recreation
- Manchester United
- Kevin Mirallas
- Roberto Martinez
- Lacina Traore
- Phil Jagielka