The 33-year-old, who spent last season on loan at Goodison Park, is a free agent after his contract at Manchester City expired earlier this summer.
Martinez has claimed that the former England international would be the perfect signing for the Toffees.
"In a World Cup summer everything goes slow and everything happens at its own pace," he told The Echo. "Gareth is such a unique person, I have been so impressed and I think everyone at the club has taken to him.
"For us he would be an important player for the future but we respect him as a human being and we are going to accept whatever he decides and make sure whatever happens we are here to support him in every aspect.
"He has been a special player for us and we will try to give him an incentive to stay even though there is nothing you can say to a player who has experienced everything in the last 10 months."
Barry scored three goals in 37 appearances for Everton last season.