England international Forster has been the subject of long-term interest from the St Mary's Stadium club, who are thought to have agreed a £10 million deal with Scottish champions Celtic.
And, speaking after his side's 1-0 friendly defeat to Bayer Leverkusen on Saturday, Southampton boss Koeman confirmed a move for the former Newcastle United man is imminent.
"Of course we are interested - he did the medical tests," the Dutchman is quoted by The Daily Echo.
"But for yesterday afternoon, I have not spoken to the board and maybe when I go up for a drink they will tell me that.
"For sure we are interested but I didn't hear he has signed."
Forster has made almost 200 appearances for Celtic in four years, earning a first England call-up in October 2012 before making his international debut against Chile in November last year.
He could be joined at Southampton by Leeds United right-back Sam Byram.
With vast financial reserves available to Koeman following the sales of Rickie Lambert, Adam Lallana, Luke Shaw, Dejan Lovren and Calum Chambers, Southampton are expected to be one of the Premier League's more active clubs in the latter weeks of the transfer window.
And with defensive reinforcements suddenly in short supply, Koeman has targeted 20-year-old Byram to boost numbers at the back following Jose Fonte recently signing an extended deal with the club.
"We are after that, we lost Chambers to Arsenal, it was a player we had on the list and it's true that we had a bid for him, yes," confirmed Koeman.
"What I heard was that Leeds didn't accept it."
Ron Vlaar has also been linked with a St Mary's switch, but reports persist over Morgan Schneiderlin and Jay Rodriguez possibly joining former Southampton manager Mauricio Pochettino at Tottenham.
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