The 24-year-old, who has figured in all three of the Toffees' Premier League matches so far this season, could be sidelined for as many as six games.
Gibson recently told Republic of Ireland manager Giovanni Trapattoni he was still suffering from the disappointment of playing a limited role at the Euro 2012 finals and did not want to be considered for selection in their World Cup qualifying campaign.
With David Moyes having let highly-rated young midfielder Ross Barkley go out on a month's loan to Sheffield Wednesday he admits his squad is slightly stretched ahead of Monday's visit of Newcastle.
"Darron's got a thigh strain and will probably be out for four to five weeks," said the Scot.
"He's had two weeks now so hopefully we are a couple of weeks down the line already because of the international break so he may have only a couple of weeks to miss. It is a concern for me because he has been very consistent and very good.
"It is a blow but we will have to find ways of getting around it. The Ross Barkley situation leaves us a bit tight in certain areas but I've taken that decision."