Clyne is out of contract in the summer of 2016 and fuelled speculation over a move to Manchester United by talking about his desire to play in the Champions League.
Koeman won the competition twice as a player and fully understands Clyne’s ambitions, and therefore the difficulty Saints may face in getting him to stay.
“It’s up to the press how they take those kind of comments,” the Dutchman said.
“I like to play Champions League as well but that’s the aspiration for every football player and it’s a normal aspiration. If you don’t mention that, there’s something wrong.
“We are still in talking with the agent of Clyney, but it’s difficult. I read in the press there’s some interest.
“Nobody knocked on the door with an offer for Clyne, but we know (he has) one season on his contract and we want Clyne to stay at the club.
“We need understanding for that but nothing else.
“We hope one day we will compare to the big ones, but that’s difficult. We know our level, we know our possibilities.
“The club can’t fight against big teams in the Premier League because the money and the possibilities of them are still much higher than we have.”
The Southampton manager says he wants Clyne’s future sorted before the end of the current season.
“We don’t wait until the end of the season because you know if there is a player who is left one season of his contract then you have to do something,” he said.
“Of course we have to do everything to keep the player in the team, but I like to know it as soon as possible – not in the last week of the season, but earlier on.
“I mentioned April but by the end of April, the first week of May, it’s good to know something about his situation.”
There is at last some positive news for Rodriguez, though, with the forward set to join up with first-team training for “some parts of the sessions” from next
“I hope that he can play a part in one of them maybe last one,” Koeman said. “Maybe it will be on one side a tough one (to return in) because it is Man City and he got this injury against them.
“That will be nice to end the season but the focus has to be next season and if he is part of one or two games this season it will be great for him, great for the team, great for the club.
“He is putting everything in to get back, but we are not in a hurry. If he comes back he has to be 100 per cent and that’s most important.”
Rodriguez’s return to fitness will bring with it heightened speculation about his future.
Like Clyne and Morgan Schneiderlin, the 25-year-old has been linked with a move away from St Mary’s, with the likes of Tottenham, Manchester City and Liverpool mentioned.
But Koeman said: “Of course I expect we don’t sell the player. The player didn’t play in my first season as a manager.
“He’s good guy, a positive one and a good player, and maybe that will be the best transfer what we can do for the next season.”
Koeman says he is proud of equalling Southampton’s Premier League points record but he has set his sights on getting more points as the club fight for European football.
“We fight for European football, that’s more important,” he told the club’s YouTube channel.
“Of course, if you take this season and they ask in the beginning about the points what we have or the clean sheets or winning games, it’s maybe more than we expect.
“But we would like to do more, to win more games and to finish in the highest position in the table.”
Koeman has not ruled out a top four finish but the Saints boss is being realistic about his team’s chances of
He said: “The door is always a little bit open, but it’s difficult. We will try.
“If we win all the remaining games, it’ll be a good chance to finish fourth – but that will be difficult.”
“At the moment, we have to focus on fifth, sixth and seventh places in the table.