The midfielder damaged his knee just before half-time during Sunday's 2-1 win over Real Betis, and joins Antolin Alcaraz on the sidelines ahead of facing Norwich City
The 25-year-old hobbled off the field just before half-time in the 2-1 win over Real Betis, which came courtesy of a Nikica Jelavic strike and a Stephan Andersen own goal.
But with the season opener against Norwich City just a week away, new manager Roberto Martinez has confirmed that Gibson may not be available.
"Darron is the only negative [from the game]," the Spaniard told the club's official website.
"We have been quite fortunate in the pre-season not to pick up too many knocks and bruises but Darron opened his knee slightly and has a grade one ligament problem.
"We will have to assess it over the next 48 hours and we’ll have a clear idea of where he is."
The Republic of Ireland international has been a regular fixture for the Premier League side since joining last year - starting 22 games - and he is not Martinez's only worry.
The 40-year-old will also be without summer signing Antolin Alcaraz, with the Paraguay international suffering with a hamstring problem.