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."
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.
"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.
