The 33-year-old spent last season on loan with the Toffees from Manchester City and manager Roberto Martinez believes the club and Barry are a "perfect fit".
Since his contract at the Etihad Stadium expired Barry has been free to talk to other clubs but it is understood the former England international is scheduled to meet Everton officials in the coming days.
The club remain optimistic of securing a player who was a key component in their success last season.
At the moment there are still other parties interested in Barry but the familiarity of Everton, in addition to their proximity which would avoid the necessity to relocate, appear to have put the Merseysiders ahead in the race.
Transfer dealings on a number of targets are taking time and have been complicated slightly by the presence of Martinez in Brazil, where he is working as television pundit.
The Spaniard will not return to England until late next week and that should see the recruitment drive pick up pace.
He always expected this to be a drawn-out summer in terms of new signings but retaining the services of Barry was near the top of his list of things to get done.
However, Martinez has given the midfielder the time and space to consider his next move rather than trying to pressure him in order to get the deal done.
"In a World Cup summer everything goes slow and everything happens at its own pace," Martinez said.
"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.
"Football happiness is a major aspect any footballer chases and Gareth and Everton are a perfect fit and the way he has been performing shows he has that footballing happiness.
"We all know what Gareth brings and how important that is."