Gibson was stretchered off during the Republic of Ireland's 3-1 World Cup qualifying win over Kazakhstan on Tuesday night.
Gibson made his international return last Friday against Germany after a spell in self-imposed exile.
He refused to play under former manager Giovanni Trapattoni, but once the Italian departed, he was brought back into the fold by interim coach Noel King, who has confirmed the news of Gibson's injury.
The setback is a big blow for Everton boss Roberto Martinez, who was in Dublin to watch Gibson, James McCarthy and Seamus Coleman all start for Ireland.
A statement on the Everton website read: "Darron Gibson is set to undergo surgery on his injured knee.
"Gibson was stretchered off during the first half of the Republic of Ireland's 3-1 win over Kazakhstan on Tuesday evening.
"A subsequent scan showed he has damaged his anterior cruciate ligament (ACL) and the midfielder will require an operation.
"The length of time the 25-year-old will be best determined once the surgery is completed.
"However, the scans did confirm to Everton's medical team that no additional damage was sustained to the structure of the right knee, ensuring all treatment and rehabilitation will be focussed on repairing the damaged ACL."
Gibson missed the start of this season and large parts of last season too because of injury.
I'm devastated, but would like to thank all the fans for their messages, long road ahead but I'll be back fitter and stronger! #COYB- Darron Gibson (@D_gibson4) October 16, 2013