According to a report in The Daily Mail last month, the Toffes are yet to offer a new deal to the Republic of Ireland international midfielder.
Former Manchester United and Leeds United midfielder Giles has suggested that Gibson should not stay at Everton next season.
The former Republic of Ireland star also believes that the 28-year-old does have an international future despite not being part of his country’s 23-man squad for the Euro 2016 finals in France this month.
“Gibson has had some bad injuries,” Giles told The Derry Journal. “He’s been out for a good while and didn’t play many games this season. I know he had a very, very serious injury but that happens. That’s part of the job. Some are lucky and some are not so lucky with injuries.
"You would like to think it’s not the end of his Ireland career. He has the summer now to get a bit of a rest. When you get one bad injury you tend to get another one and that’s happened to him.
"So he should get his body back to the way it should be and maybe a move away from Everton might help as well.”
Gibson managed to make just seven appearances in the Premier League for Everton this past season, creating only four chances in the process.
The former Manchester United midfielder had a pass accuracy of 85% and a shot accuracy of 50%, and won seven headers and three dribbles during the 2015-16 campaign.