Koeman's men have been the surprise package of 2014-15 in the Premier League and sit second after 10 games going into Saturday's home clash with Leicester City.
The challenge now facing Southampton is to continue to exceed expectations and the Dutchman - who succeeded Mauricio Pochettino in June - believes a place in next season's UEFA Europa League is a realistic aim for the south-coast club.
He said: "We would like to continue in the good direction, like the club is heading in the past three-four years.
"Of course the next step is playing in Europe, we know that it's a dream for the club, it's a dream for the fans and we have to do it.
"I have confidence that we can reach that, then we have to keep our ambition and spirit in the team and that's the most important thing we have to bring on a daily basis."
Despite being just our points behind leaders Chelsea, Koeman is not entertaining the idea of a title challenge for Southampton.
The former Feyenoord manager coach said: "I think it's a two-team race because still I believe that [Manchester] City and Chelsea are stronger over the whole season, not in one game because I think we can beat them.
"But on the whole season I believe they are stronger than us. We know our situation we know our ambition but we live game by game and it's not important or necessary to look more forward."
Southampton unveiled a new state-of-the-art training centre on Wednesday as they look to continue their long success of nurturing exciting young talent.
Five academy graduates - including Arsenal's Theo Walcott - were named in Roy Hodgson's England squad on Thursday for upcoming games against Scotland and Slovenia and Koeman believes it is no surprise.
He said: "That means a lot to the club, it means we have a lot of qualities in academy. We know that, the club Southampton is famous for that."
