Barry, 35, believes the changes new manager Ronald Koeman has made will make them more competitive after being criticised in predecessor Roberto Martinez's last two seasons for being too open.
"The manager made a point in terms of the way he wants the training to change and I could see where he was coming from," said Barry.
"He likes the hard work to be put in and that's fair enough. The training is a bit more structured and we're doing a lot more afternoon sessions to previous.
"His standards are really high; he wants us to keep them and as soon as anybody drops them, no matter who you are, he lets you know so it's good for all the players involved.
"So seven points from nine, and through to the next round in the League Cup, it's a great start.
"The club just wants to be playing European football regularly. It's a huge club and the fans deserve to be seeing European nights every season so there's no reason why we can't be targeting those places."