Clyne has impressed at the St Mary's Stadium this season and his form has seen him earn international recognition with England.
The former Crystal Palace full-back has subsequently been linked to several suitors, with the likes of Manchester United, Chelsea and Arsenal thought to be interested.
"We are still trying to keep Clyne in the club. We know it's not easy as maybe there is some interest," Koeman told BBC Radio Solent.
"At the moment, there is not a serious bid for whichever player in the team, but we'll do everything to keep them."
Koeman is also seeking clarity on the future of on-loan Atletico Madrid defender Toby Alderweireld.
The Liga champions' sporting director Jose Luis Perez recently stated that the Belgian would return to the Spanish capital next season.
"Always, you like to have an answer as soon as possible," he added. "It's more the business of Les Reed [Southampton's director of football].
"We know he wants to be at Southampton. I've read a comment that the technical director at Atletico Madrid wants him to return, so we will know in some weeks what will happen with Toby."
