Forster has started all three Premier League fixtures since his arrival from Celtic, but despite acknowledging that the England international is his number-one choice, Koeman has called on Boruc to fight for his place at St Mary's Stadium.
The 51-year-old is quoted by Sky Sports News as saying: "I've had a good conversation with Artur. Of course he was a little bit disappointed about the future, because we spoke about that and the best for him was playing in another team.
"Finally he's a player for Southampton and he knows we start again, we start like we did at the beginning of the season.
"If he shows the spirit and ambition that we expect from every player, he can always change his situation."
Southampton play host to Newcastle United on Saturday afternoon.