The Scottish Under-21 keeper will play in the club's first pre-season friendly on Saturday against FC Sion in Switzerland.
Turner, and youngster John Ruddy, are the only two keepers attending Moyes' training camp.
Turner missed most of last season with a serious hip injury but has worked hard during the summer to regain fitness to give him the chance to stake his claim at Goodison Park.
He said: "Last season was a write-off for me. I have been in a lot of the summer trying to get my fitness back so at the moment I am feeling good.
"My fitness is getting there but it has been a hard couple of weeks."
That extended spell on the sidelines has made the goalkeeper all the more determined to make headway at Everton.
He added: "All long-term injuries are hellish, I just want to be playing week-in, week-out. I just want to get fit and get back in the team.
"I am dying for a game. During the whole of last season I played about three-and-a-half games, so I just want to get matches under my belt.
"I just want get back in the team, prove myself and show everyone what I can do.
"Everyone wants to be number one for their club, number one for their country. That is every goalkeeper's dream."