Gibson sat out the Toffees' 2-1 home defeat to Chelsea on Sunday with the injury, and Moyes has since revealed the Republic of Ireland international will be absent for "about four weeks".
|11/1||Everton are 11/1 with Bet365 to lose 1-0 at Newcastle
The 25-year-old only returned to first-team action in his side's 1-1 draw with Arsenal at the end of November, having missed almost two whole months of the campaign with a similar problem.
When asked about the knock, Moyes told reporters: "I think he's going to be out for about four weeks."
This latest setback looks likely to rule Gibson out of Everton's Premier League clashes with Newcastle, Swansea, Southampton and West Brom, as well as an FA Cup Third Round tie with Cheltenham.
The Toffees are already without first-teamers Kevin Mirallas, Tony Hibbert and Seamus Coleman through injury, though star man Marouane Fellaini will return from suspension for Wednesday's clash with the Magpies.