Koeman recently cast doubt on Clyne's future by stating that Southampton face a battle to keep the England international amid reported interest from Chelsea and Manchester United.
Clyne subsequently conducted an interview with Sky Sports earlier this week where he spoke of his interest of playing in Europe's premier competition.
However, Koeman says that it should be the goal for any professional footballer to play on the biggest stage.
"It's up to the press how they take that kind of comments. I would like to play in the Champions League as well, it's the aspiration for every player, it's a normal aspiration," he said.
"If you don't mention that there's something wrong. We're discussing with his agent to keep Clyney at the club. We know it's difficult because I read in the press that there's interest."
Southampton are not out of the race for the Champions League themselves and sit sixth, just five points adrift of fourth-placed Manchester City.
Asked if qualifying for the tournament would convince Clyne to stay, Koeman added: "Maybe that can be an influence in his final decision.
"I think always players need to have aspirations. We hope one day we'll compare to the big ones."
Champions City have lost three of their past four league matches, but Koeman believes Manuel Pellegrini's men remain favourites for fourth.
"The door is always a little bit open. It's difficult because even if Man City have been struggling in recent weeks with results they have now two home games and they have enough quality to be in the top four," the Dutchman said.
"If we win all of [our games] it's a big chance to finish fourth in the league, but to win all the last six games is difficult."
