Clyne has been one of Saints' stand-out performers this season and has enhanced his growing reputation this week with impressive displays for England against Slovenia and Scotland.
The 23-year-old's contract will expire in the summer of 2016 and Southampton are eager to tie him down before the deal runs into its final year.
"I know Clyney's contract has one year left after this season so it's normal to do something now to keep that player," Koeman said.
"Have those talks already started? Of course, you cannot wait until next season."
He continued: "I have had conversations with (executive director) Les Reed about the contracts of the players.
"We want to keep everyone and I am not surprised about Clyne's performances for the national team.
"He is a fantastic player which he has showed since the start of the season.
"If there is interest, we will hear about that and we will try everything to keep him here because he is fantastic football player."
Southampton spent most of last summer fielding enquiries for their talented squad and another player set to spark interest again in January is Morgan Schneiderlin.
The midfielder appeared non-committal on his Saints future when he told French newspaper L'Equipe during the international break that he wanted to play for a "big club".
"I say nothing about his future because he is a Southampton player," Koeman said.
"I also know - I had that experience as a player - that when you go back to play for your national team you do interviews and they make it a bit different and then it comes back to England and it is a big story.
"He is a Southampton player and he is focused on Aston Villa on Monday."
Schneiderlin returned from international duty with a groin injury but Koeman is hopeful the 25-year-old will be available for the clash at Villa Park.
Steven Davis is a doubt with a hamstring problem sustained against Leicester but Dusan Tadic is fit again.
Southampton face a difficult Christmas period with games against Manchester City, Arsenal, Manchester United, Everton and Chelsea all to come before the end of the year.
Koeman, however, insists his team will not change their style of play against the league's tougher opponents.
"It's always easier to keep winning when the start is positive," Koeman said.
"The belief the players have is important to win games and in my opinion the most important thing is we keep our way of playing.
"Even if we play away or home at Manchester City or Liverpool, that has to be the same."
Source : PA
Source: PA