Kevin Mirallas, Steven Naismith and Aiden McGeady are all reportedly considering their futures at Everton.
Gibson is keen to play at Euro 2016 with the Republic of Ireland, but he realises that his chances of being selected are limited unless he can play regular first team football.
Those chances look unlikely to come at Goodison Park, as he has slipped some way down the pecking order in Roberto Martinez’s team.
The likes of Gareth Barry, Tom Cleverley, Muhamed Besic and James McCarthy are all ahead of Gibson in the pecking order at the Merseyside outfit.
Gibson has only played four times for Everton this term, and he is now starting to think about his future.
Martinez refused to rule out the departures of players in January at his press conference yesterday, as covered by the club's official YouTube channel.
As well as Gibson, Aiden McGeady is also said to be on the verge of leaving Everton.
The flamboyant winger has been starved of first team opportunities, and is keen to move on in the new year, the Daily Post report.