Gibson had started all three of Everton's Premier League games so far this season, but his injury means he could miss their next six matches. The 24-year-old, who joined Everton from Manchester United in January for an undisclosed fee last season, suffered the problem in the first half of Everton's 2-0 defeat at West Bromwich Albion before the international break.
"Darron's got a thigh strain and will probably be out for four to five weeks now," Moyes said. "He's had two weeks now so hopefully we are a couple of weeks down the line already because of the international break."
Gibson recently told Republic of Ireland manager Giovanni Trapattoni he was still suffering from the disappointment of playing a limited role at UEFA EURO 2012 and did not want to be considered for selection in their 2014 FIFA World Cup™ qualifying campaign.