The experienced midfielder moved to Goodison Park on a season-long loan after being deemed surplus to requirements at Manchester City.
With his contract at the Etihad Stadium due to expire at the end of the current campaign, Barry is free to discuss his future with prospective employers from January.
The 32-year-old concedes he is entering the unknown in many ways, but admits he would be more than happy to discuss an extended stay on Merseyside.
Barry told Everton's official website: "Personally, it's a really strange situation, I've never had it before.
"In January you can start discussing your future for next season but I think the club made it clear that they would probably like to discuss things at the end of the season and if I put in fantastic performances from here until then, then that puts me in a good position.
"I'm really enjoying it and I'm as excited now as when I signed because I've got a feel for the club, I'm enjoying working under the manager and the coaching staff here, the training is fantastic and the style of play the manager wants us to play is really enjoyable, so I'm really excited for the rest of the season.
"A lot of people have spoken to me and said you could have so many options going into your last year but, for me, now it's just about playing for Everton, keeping a good run of form going and taking it from there."
Next up for Everton is a trip to Premier League leaders Arsenal on Sunday - a club Barry admits expressed an interest in acquiring his services once it became clear that City were prepared to let him leave.
He added: "There was a bit of contact with Arsenal and they were interested, but I think they went down the road with (Mathieu) Flamini.
"If I'm being honest, they have sort of surprised me, the way they're playing, the way they're being relentless in winning games. Their style of football hasn't changed but some of their players are at the top of their form.
"But the lads in the dressing room can go down to Arsenal now and play with no fear, believing we can put in another performance like we did at Old Trafford. The win there takes a lot of pressure off us and we'll be going down to London looking to do a similar sort of job."