The Republic of Ireland international missed Sunday's 2-1 home defeat by Chelsea and afterwards manager David Moyes said he expected the player to be out for a while.
On Monday afternoon Moyes was able to put a more specific timescale on Gibson's absence.
"I think he's going to be out for about four weeks," said the Scot.
The 25-year-old spent more than two months on the sidelines earlier in the season with a similar problem.
He missed 10 Barclays Premier League matches as a result and a further absence is a significant blow to Everton as he has been one of their key players since arriving in January.
Another injury is the last thing Moyes needed as he still has Kevin Mirallas (hamstring), Tony Hibbert (calf) and Seamus Coleman out, while Phil Neville (knee) had to be rested on Sunday having played two matches in five days on his comeback from surgery.