Barkley spent the 2020-21 season on loan at Aston Villa from Chelsea.
The midfielder now returns in a free transfer for an undislosed length of contract.
"For me, and everyone else at the club and the fans, (the ambition) is to win a trophy,” he said.
“The club came so close last season and were unfortunate. With additions to the already great squad that we’ve got it’s going to be pivotal for the season. Hopefully we can win a trophy.”
Barkley added: “Over the last few years the club has made huge strides since I was last at the club.
“As an outsider looking in, it looked fantastic, and to be a part of that is fantastic for me.
“Playing in the Champions League is a big incentive for me and the club, so I’m looking forward to that.
"I know plenty of staff and I know plenty of the players as well.
“I’m really close to John McGinn and I speak to him often, Ezri as well. I’m looking forward to getting back in.
“I’m in a week early now and staying on top of things. I’m looking forward to meeting everybody when they return.”
Barkley added: “The last time I came, I didn’t play in front of the fans, so I’m looking forward to that.
“The atmosphere is great, so I’m looking forward to that as well as the Champions League nights. It’s been a long time since the club were in the Champions League, so it’s going to be a massive season.”